The Mathematics of Maybe: Introduction to Probability Notes

What is Probability?

Probability is a mathematical concept used to measure the likelihood of an event occurring. It quantifies our uncertainty about the outcome of an event, expressing how confident we are that it will happen. Instead of measuring physical quantities like length or volume, probability measures the chance of an event.

For example, questions like "Will it rain today?", "Will our school win the hockey match?", or "Will I be selected for the lucky draw?" involve situations where we know the possible outcomes, but we cannot predict with certainty which outcome will actually occur. These are known as random events.

What is Randomness?

Randomness describes situations or actions where the outcome cannot be precisely predicted in advance, even though all possible outcomes are known. This unpredictability is often referred to as an element of chance.

Tossing a coin: You know the possible outcomes are heads or tails, but you cannot be sure which will appear on a single toss.
Rolling a die: The possible outcomes are the numbers 1, 2, 3, 4, 5, or 6. However, you cannot know which number will land face up on any given roll.

These unpredictable occurrences are called random experiments or trials in probability. The fairness of a coin toss for deciding which team bats first in cricket is a prime example of how randomness can be useful.

Probability is the branch of mathematics that studies randomness and the likelihood of specific outcomes in random situations. For instance, in a coin toss, the probability of getting heads is $\frac{1}{2}$ and the probability of getting tails is $\frac{1}{2}$ , because each outcome is equally likely.

Weather events like rain are considered random because they depend on numerous complex atmospheric factors (temperature, humidity, wind, pressure) that are so sensitive that perfect prediction is impossible. While the exact timing and location of rain are unpredictable, we can estimate its probability using patterns and data.

Example of Randomness

Given: You are asked to predict the outcome of a coin toss. To find: Whether you can know the outcome with certainty. Solution:

When a coin is tossed, the possible outcomes are heads (H) or tails (T).
There is no way to know in advance whether the coin will land on heads or tails for a single toss.
Your friend might guess heads or tails, but this is a prediction, not a certainty. Answer: The outcome of a coin toss is unpredictable, demonstrating randomness.

The Probability Scale

Probability is measured on a scale from 0 to 1. This scale helps us understand and compare the likelihood of different events:

A probability of 0 means the event is impossible.
A probability of 1 means the event is certain.
Probabilities between 0 and 1 indicate varying degrees of likelihood.
A probability of 0.5 signifies an equally likely chance, meaning the event is as likely to happen as it is not to happen (like flipping a coin and getting heads).
A probability greater than 0.5 indicates the event is more likely to occur.
A probability less than 0.5 indicates the event is less likely to occur.

The scale can also be represented using percentages: 0% for impossible, 100% for certain, and 50% for equally likely.

Example Illustrating the Probability Scale

Imagine a deck of six cards where the number of purple and green cards is unknown. The probability of picking a purple card can range from:

Impossible (0): If there are no purple cards.
Certain (1): If all six cards are purple.
As the number of purple cards increases, the likelihood of selecting a purple card moves from less likely, to even chance, to more likely, along the probability scale.

Here's how different events can be ranked on this scale:

Event	Likelihood Category	Reason
Getting a number greater than 6 on a die	Impossible	A standard die only has numbers from 1 to 6.
Rolling a 3 on a standard die	Less likely	While not impossible, only one face of the die shows a 3.
Flipping a coin and getting heads	Even chance	Heads and tails are equally likely outcomes.
Drawing any number from 2 to 10 from a deck of 52 cards	More likely	Many cards in a standard deck have numbers between 2 and 10.
Choosing a red sweet from a bag of all red sweets	Certain	If all sweets in the bag are red, picking a red one is guaranteed.

Example Problem on Probability Scale

Given: A scenario where an event's likelihood needs to be ranked. To find: The ranking of given events on a scale from 0 (Impossible) to 1 (Certain), with reasons.

Ranking the events:

(i) The next Monday will come after Sunday.

Rank: Certain (1)
Reason: The sequence of days of the week is fixed and follows a regular pattern.

(ii) It will snow in Mumbai in July.

Rank: Impossible (0)
Reason: July is a monsoon month in Mumbai, and temperatures are too high for snowfall.

(iii) An elephant will walk through your classroom today.

Rank: Impossible (0)
Reason: Elephants are not typically found in classrooms; this is an extremely unlikely event.

(iv) You will greet at least one friend at school tomorrow.

Rank: More likely (closer to 1 than 0.5)
Reason: Assuming you have friends at school, it is highly probable that you will interact with at least one of them.

Answer: (i) Certain, (ii) Impossible, (iii) Impossible, (iv) More likely.

Remember
The probability scale always ranges from 0 (impossible) to 1 (certain). Events with probabilities closer to 1 are more likely, and those closer to 0 are less likely.

Exam-smart
Case/source/data questions might present you with a situation and ask you to estimate the probability of an event using terms like "impossible," "likely," or "certain." You will need to reason based on the information given, similar to ranking the events in the example. For instance, a question might describe weather patterns and ask about the likelihood of rain.

Measuring Probability Objectively

Objective methods for estimating probability rely on evidence. This evidence comes from either performing experiments repeatedly or analysing statistical data from past observations. When we use this evidence, we calculate the relative frequency of an event. This calculated probability is known as experimental probability.

Experimental Probability: Performing Observations or Experiments

Experimental probability is determined by conducting an experiment and observing its outcomes.

An outcome is a single result of an experiment.
The sample space is the set of all possible outcomes of an experiment.

Example 1: Sample Spaces

Experiment: Tossing a coin
Possible Outcomes: Heads (H), Tails (T)
Sample Space: {H, T}
Experiment: Rolling a standard six-sided die
Possible Outcomes: The numbers 1, 2, 3, 4, 5, or 6
Sample Space: {1, 2, 3, 4, 5, 6}

The formula for experimental probability is:

P (Event) = \frac{Number of times the event occurred}{Total number of trials}

This is also referred to as the relative frequency of the event. It's particularly useful when dealing with real-world data where theoretical predictions might not perfectly match actual results.

Example

Given: A die is rolled 50 times, and the number 4 appears on the top face 8 times. To find: The experimental probability of rolling a 4.

Solution: The event is rolling a 4. Number of times the event occurred = 8 Total number of trials = 50 Using the formula for experimental probability:

P (rolling a 4) = \frac{Number of times 4 occurred}{Total number of rolls}

P (rolling a 4) = \frac{8}{50}

Simplify the fraction:

P (rolling a 4) = \frac{4}{25}

Convert to a decimal:

P (rolling a 4) = 0.16

Convert to a percentage:

P (rolling a 4) = 16 %

Answer: The experimental probability of rolling a 4 is $\frac{4}{25}$ , or 0.16, or 16%.

Example

Given: In a survey of 200 people, 130 preferred tea over coffee. To find: The experimental probability that a randomly selected person prefers tea.

Solution: The event is a person preferring tea. Number of people who preferred tea = 130 Total number of people surveyed = 200 Using the formula for experimental probability:

P (preferring tea) = \frac{Number of people who prefer tea}{Total number of people surveyed}

P (preferring tea) = \frac{130}{200}

Simplify the fraction:

P (preferring tea) = \frac{13}{20}

Convert to a decimal:

P (preferring tea) = 0.65

Convert to a percentage:

P (preferring tea) = 65 %

Answer: The experimental probability that a randomly selected person prefers tea is $\frac{13}{20}$ , or 0.65, or 65%.

Remember
Experimental probability is based on what actually happens when you perform an experiment or observe data. The more trials you conduct, the closer your experimental probability is likely to get to the true probability of the event.

Exam-smart
You might be given a table of results from an experiment (like coin tosses or dice rolls) and asked to calculate the probability of a specific outcome. You'll use the formula: $\frac{Frequency of the outcome}{Total number of trials}$ .

Theoretical Probability

Theoretical probability is based on reasoning about all possible outcomes when there's no reason to believe one outcome is more likely than another. It assumes an ideal, fair situation without needing to perform experiments.

The formula for theoretical probability is:

P (Event) = \frac{Number of favourable outcomes}{Total number of possible outcomes}

This approach is used when all outcomes are equally likely.

Example

Given: A fair six-sided die is rolled. To find: The theoretical probability of rolling a 5.

Solution: The sample space is {1, 2, 3, 4, 5, 6}. Total number of possible outcomes = 6. The favourable outcome is rolling a 5. Number of favourable outcomes = 1. Using the formula for theoretical probability:

P (rolling a 5) = \frac{Number of favourable outcomes}{Total number of possible outcomes}

P (rolling a 5) = \frac{1}{6}

Answer: The theoretical probability of rolling a 5 is $\frac{1}{6}$ .

Example

Given: A bag contains 5 red balls and 3 blue balls. All balls are of the same size and shape. A ball is drawn at random from the bag. To find: (i) The theoretical probability of drawing a red ball. (ii) The theoretical probability of drawing a blue ball.

Solution: Total number of balls in the bag = Number of red balls + Number of blue balls = 5 + 3 = 8. So, the total number of possible outcomes is 8.

(i) For drawing a red ball: Number of favourable outcomes (red balls) = 5. Using the formula for theoretical probability:

P (red ball) = \frac{Number of red balls}{Total number of balls}

P (red ball) = \frac{5}{8}

(ii) For drawing a blue ball: Number of favourable outcomes (blue balls) = 3. Using the formula for theoretical probability:

P (blue ball) = \frac{Number of blue balls}{Total number of balls}

P (blue ball) = \frac{3}{8}

Answer: (i) The theoretical probability of drawing a red ball is $\frac{5}{8}$ . (ii) The theoretical probability of drawing a blue ball is $\frac{3}{8}$ .

Remember
Theoretical probability tells us what should happen in an ideal scenario. It's calculated before any experiment is done.

Exam-smart
For theoretical probability, you'll need to identify the total number of possible outcomes and the number of outcomes that satisfy the event you're interested in. For example, if asked about the probability of getting an even number when rolling a die, the favourable outcomes are {2, 4, 6} (3 of them), and the total outcomes are {1, 2, 3, 4, 5, 6} (6 of them), so the probability is $\frac{3}{6} = \frac{1}{2}$ .

Try this: A card is drawn from a well-shuffled deck of 52 playing cards. What is the theoretical probability of drawing an Ace? Answer: $\frac{4}{52}$ or $\frac{1}{13}$

Theoretical Probability

Theoretical probability is the likelihood of an event occurring based on mathematical reasoning, assuming all possible outcomes are equally likely. It's what we expect to happen in an ideal, perfectly fair situation, without needing to perform any experiments or collect data.

The formula for theoretical probability is:

P (Event) = \frac{Number of favourable outcomes}{Number of possible outcomes}

Where:

Favourable outcomes are the specific results we are interested in.
Possible outcomes are all the potential results that could occur.

Analysing Statistical Data Using Probability

While theoretical probability deals with ideal scenarios, statistical probability uses collected data to estimate the likelihood of events. This is incredibly useful in real-world applications like business forecasting, insurance, and scientific research.

When we use statistical probability, we are essentially using data from a sample (a smaller group) to make educated guesses about a larger population (the entire group). The more representative and larger the sample, the more accurate our statistical probability estimates will be.

This process of using a sample to understand a population is called sampling.

Example 5: Favourite Fruit Survey

Given: A survey of 50 students revealed their favourite fruits: 20 like mango, 15 like apples, 10 like bananas, and 5 like grapes.

To find: The probability that a randomly picked student's favourite fruit is mango.

Solution:

Identify the number of favourable outcomes (students who like mango): 20
Identify the total number of possible outcomes (total students surveyed): 50
Apply the probability formula:

P (Mango) = \frac{Number of students who like mango}{Total number of students}

P (Mango) = \frac{20}{50}

P (Mango) = \frac{2}{5} = 0.4

Answer: The probability that a randomly picked student's favourite fruit is mango is 0.4 or 40%.

Now, if we want to estimate fruit purchases for a school of 1500 students based on this sample:

Estimated mangoes needed = 0.4 $\times$ 1500 = 600 mangoes.

To improve confidence in such estimates, a larger and more representative sample is chosen.

Remember
Theoretical probability assumes an ideal, fair scenario. Statistical probability uses real-world data, so results are estimates and can vary.

Exam-smart
Case studies might present data from a survey and ask you to calculate the probability of a specific outcome (like a student preferring a certain club) or to estimate the number of people in a larger population who might have that preference. You'll need to identify favourable and total outcomes from the given data.

Elements of Probability: Sample Spaces and Events

This section introduces the foundational concepts of sample spaces and events in probability. Understanding these is crucial for analysing random phenomena and calculating the likelihood of specific outcomes.

Sample Space

The sample space (denoted by $S$ ) is the complete list of all possible outcomes of a random experiment. Each individual outcome within this list is called an element or a member of the sample space.

Key Properties of a Sample Space:

It must encompass every single possible outcome.
Each outcome should be listed only once.
The sample size, denoted by $n (S)$ , is the total number of elements (outcomes) in the sample space.

Examples of Sample Spaces:

Experiment	Sample Space (S)	Sample Size (n(S))	Why it Matters
Raining tomorrow	{Rain, No Rain}	2	Simplest way to represent a binary outcome.
A football match outcome	{Win, Lose, Draw}	3	Covers all distinct results for a team.
Tossing a fair coin once	{Heads (H), Tails (T)}	2	Defines the two unique results of a coin toss.
Rolling a 6-sided die once	{1, 2, 3, 4, 5, 6}	6	Lists every face value possible on a standard die.
Tossing two fair coins	{HH, HT, TH, TT}	4	Lists all combinations of outcomes for two coins.
Picking fruit (Apple, Banana, Orange)	{Apple, Banana, Orange}	3	Lists the distinct types of fruit available to pick.

Important Note on Detail: The level of detail in a sample space depends on the specific problem. For instance, if differentiating between light rain and heavy rain is important, the sample space for rain would need to be expanded.

Example 1: Rolling a Die and Tossing a Coin

Given: An experiment involves rolling a 6-sided die and tossing a coin. To find: The sample space $S$ and the sample size $n (S)$ . Solution:

Possible outcomes for the die roll: {1, 2, 3, 4, 5, 6}
Possible outcomes for the coin toss: {H, T}
To find all combined outcomes, we pair each die outcome with each coin outcome.
1 with H: 1H
2 with H: 2H
3 with H: 3H
4 with H: 4H
5 with H: 5H
6 with H: 6H
1 with T: 1T
2 with T: 2T
3 with T: 3T
4 with T: 4T
5 with T: 5T
6 with T: 6T
$S = {1 H, 2 H, 3 H, 4 H, 5 H, 6 H, 1 T, 2 T, 3 T, 4 T, 5 T, 6 T}$
$n (S) = 12$

Answer: $S = {1 H, 2 H, 3 H, 4 H, 5 H, 6 H, 1 T, 2 T, 3 T, 4 T, 5 T, 6 T}$ , $n (S) = 12$

Example 2: Choosing an Integer

Given: Choosing a random integer between – 5 and + 5 (inclusive). To find: The sample space $S$ and the sample size $n (S)$ . Solution:

The integers between – 5 and + 5 are: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5.
$S = {- 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5}$
To count $n (S)$ : (Last Number - First Number) + 1 = $(5 - (- 5)) + 1 = (5 + 5) + 1 = 10 + 1 = 11$ .
$n (S) = 11$

Answer: $S = {- 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5}$ , $n (S) = 11$

Example 3: Balls in a Box

Given: A box contains 5 green balls and 7 red balls. One ball is drawn at random. To find: The sample space $S$ and the sample size $n (S)$ . Solution:

When one ball is drawn, the outcome is the colour of the ball.
The possible colours are Green (G) and Red (R).
$S = {G, R}$
$n (S) = 2$ (We are interested in the type of outcome, not the individual balls if they are indistinguishable by colour.)

Answer: $S = {G, R}$ , $n (S) = 2$

Example 4: Snack and Drink Combinations

Given: 3 snacks (Samosa, Pakora, Bhaji) and 2 drinks (Chai, Lassi). To find: The sample space $S$ of snack and drink combinations. Solution:

We pair each snack with each drink.
Samosa with Chai: (Samosa, Chai)
Samosa with Lassi: (Samosa, Lassi)
Pakora with Chai: (Pakora, Chai)
Pakora with Lassi: (Pakora, Lassi)
Bhaji with Chai: (Bhaji, Chai)
Bhaji with Lassi: (Bhaji, Lassi)
$S = {(S a m o s a, C h a i), (S a m o s a, L a s s i), (P a k o r a, C h a i), (P a k o r a, L a s s i), (B h a j i, C h a i), (B h a j i, L a s s i)}$
The sample size $n (S)$ is $3 snacks \times 2 drinks = 6$ .

Answer: $S = {(S a m o s a, C h a i), (S a m o s a, L a s s i), (P a k o r a, C h a i), (P a k o r a, L a s s i), (B h a j i, C h a i), (B h a j i, L a s s i)}$

Events

An event is a specific outcome or a collection of outcomes from the sample space. It represents a particular result or set of results that we are interested in. Events are essentially subsets of the sample space.

How Events Relate to Sample Spaces:

You perform a random experiment.
You identify all possible outcomes, which form the sample space ( $S$ ).
You then identify a specific set of outcomes that you are interested in; this set is called an event ( $E$ ).

Examples of Events:

Experiment: Tossing Two Coins

Sample Space: $S = {H H, H T, T H, T T}$
Event E: ‘At least one coin shows Head.’
This event includes outcomes where H appears once or twice.
$E = {H H, H T, T H}$
The number of outcomes in this event is $n (E) = 3$ .

Experiment: Rolling a 6-sided Die

Sample Space: $S = {1, 2, 3, 4, 5, 6}$
Event E: ‘The number rolled is greater than 4.’
The numbers in the sample space greater than 4 are 5 and 6.
$E = {5, 6}$
The number of outcomes in this event is $n (E) = 2$ .

Experiment: Picking Fruit from a Basket (Apple, Banana, Orange)

Sample Space: $S = {A p p l e, B a n a n a, O r a n g e}$
Event E: ‘Picking a fruit that is yellow.’
Assuming Bananas are yellow.
$E = {B a n a n a}$
The number of outcomes in this event is $n (E) = 1$ .

Experiment: Snack and Drink Combinations

Sample Space: $S = {(S a m o s a, C h a i), (S a m o s a, L a s s i), (P a k o r a, C h a i), (P a k o r a, L a s s i), (B h a j i, C h a i), (B h a j i, L a s s i)}$
Event E: ‘Selecting Samosa as a snack.’
This event includes all combinations where Samosa is the snack.
$E = {(S a m o s a, C h a i), (S a m o s a, L a s s i)}$
The number of outcomes in this event is $n (E) = 2$ .

Example 5: Rolling a Die (Specific Event)

Given: A single 6-sided die is rolled. To find: The sample space $S$ and the event $E$ of rolling an even number. Solution:

The sample space for rolling a die is $S = {1, 2, 3, 4, 5, 6}$ .
The even numbers in this sample space are 2, 4, and 6.
The event $E$ of rolling an even number is $E = {2, 4, 6}$ .
The sample size is $n (S) = 6$ .
The number of outcomes in the event is $n (E) = 3$ .

Answer: $S = {1, 2, 3, 4, 5, 6}$ , $E = {2, 4, 6}$

Example 6: Tossing Two Coins (Specific Event)

Given: Two fair coins are tossed simultaneously. To find: The sample space $S$ and the event $F$ of getting exactly two tails. Solution:

The sample space for tossing two coins is $S = {H H, H T, T H, T T}$ .
The outcome with exactly two tails is TT.
The event $F$ of getting exactly two tails is $F = {T T}$ .
The sample size is $n (S) = 4$ .
The number of outcomes in the event is $n (F) = 1$ .

Answer: $S = {H H, H T, T H, T T}$ , $F = {T T}$

Remember
A sample space must list all possible outcomes, while an event is a specific collection of outcomes from that sample space that we are interested in.

Exam-smart
In a case study, you might be given a scenario (like drawing cards from a deck or spinning a spinner) and asked to list the sample space or identify a specific event based on certain conditions. You need to be able to systematically list all possibilities and then pick out the ones that match the event description. For instance, if a question describes spinning a spinner with 5 sections numbered 1 to 5, the sample space is $S = {1, 2, 3, 4, 5}$ . An event like "the spinner lands on an odd number" would be $E = {1, 3, 5}$ .

Try this: A box contains 3 red balls, 4 blue balls, and 2 yellow balls. One ball is drawn at random. List the sample space and the event of drawing a blue ball. Answer: $S = {R, B, Y}$ , Event $E = {B}$

Tree Diagrams

A tree diagram is a visual tool used to represent and list all possible outcomes of a multi-step experiment. A multi-step experiment is one that involves a sequence of separate events or trials. Each branch on the tree represents a possible outcome of an individual step, and these branches split further to show the different paths and outcomes for subsequent events.

Tree diagrams are particularly useful for:

Visualising complex experiments where the order of events matters.
Listing all possible outcomes in a sample space systematically.
Calculating the theoretical probability of combined events.

Visualising Multi-Step Experiments

Let's consider an experiment of tossing a fair coin two times. This is a multi-step experiment because there are two distinct tosses.

Step 1: First Toss The first toss can result in either Heads (H) or Tails (T). We represent these as branches from a starting point.
Step 2: Second Toss For each outcome of the first toss, there are two possible outcomes for the second toss (H or T). So, from the end of the 'H' branch, we draw two new branches for the second toss (H and T). Similarly, from the end of the 'T' branch, we draw two new branches for the second toss (H and T).

Each path from the start of the diagram to an end point represents a complete outcome of the experiment.

Example 1: Tossing a Fair Coin Twice

Given: An experiment involving tossing a fair coin two times.

To find: The sample space and the probabilities of each outcome using a tree diagram.

Solution:

Start with a single point.
Draw two branches representing the outcomes of the first toss: H (Heads) and T (Tails).
From the end of the H branch, draw two more branches for the second toss: H and T. These represent the outcomes HH and HT.
From the end of the T branch, draw two more branches for the second toss: H and T. These represent the outcomes TH and TT.

The tree diagram shows four possible paths from start to end.

(Start) /
H T (First Toss) / \ /
H T H T (Second Toss)

The complete outcomes are:

HH (Heads on first toss, Heads on second toss)
HT (Heads on first toss, Tails on second toss)
TH (Tails on first toss, Heads on second toss)
TT (Tails on first toss, Tails on second toss)

The sample space $S$ is the set of all possible outcomes: $S = {H H, H T, T H, T T}$ .

Since the coin is fair, each of these 4 outcomes is equally likely. The theoretical probability of each outcome is calculated as:

P (Outcome) = \frac{Number of favourable outcomes}{Total number of possible outcomes}

For any specific outcome like HH, there is 1 favourable outcome out of 4 total.

P (H H) = \frac{1}{4} = 0.25 or 25 %

Similarly, $P (H T) = \frac{1}{4}$ , $P (T H) = \frac{1}{4}$ , and $P (T T) = \frac{1}{4}$ .

Answer: The sample space is ${H H, H T, T H, T T}$ , and the probability of each outcome is $\frac{1}{4}$ .

Calculating Probability of Combined Events

Tree diagrams can also be used to calculate the probability of events that involve a sequence of outcomes. The probability of a specific path in a tree diagram is found by multiplying the probabilities of each branch along that path.

Let's consider the event of getting exactly one head and one tail in two coin tosses. This can happen in two ways: HT or TH.

Probability of HT:

P (H on 1st toss) = \frac{1}{2}

P (T on 2nd toss) = \frac{1}{2}

P (H T) = P (H on 1st) \times P (T on 2nd) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

Probability of TH:

P (T on 1st toss) = \frac{1}{2}

P (H on 2nd toss) = \frac{1}{2}

P (T H) = P (T on 1st) \times P (H on 2nd) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

Since the events HT and TH are mutually exclusive (they cannot happen at the same time), the probability of getting one head and one tail is the sum of their probabilities:

P (One Head and One Tail) = P (H T) + P (T H) = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}

Example 2: Picking Fruits from Two Baskets

Given: Basket A has 1 apple (A) and 2 oranges (O). Basket B has 1 banana (B) and 1 mango (M). You pick one fruit from each basket.

To find: (i) A tree diagram showing all possible pairs of fruits. (ii) The sample space. (iii) The probability of picking one apple and one banana.

Solution: (i) Tree Diagram:

First, consider picking from Basket A. The outcomes are A and O.
From the 'A' outcome in Basket A, draw branches for the outcomes from Basket B (B and M). This gives the combinations AB and AM.
From the 'O' outcome in Basket A, draw branches for the outcomes from Basket B (B and M). This gives the combinations OB and OM.

Basket A Basket B Combinations (A) ----- (B) -> AB | | | -(M) -> AM | (O) ----- (B) -> OB | -(M) -> OM

(ii) Sample Space: The sample space $S$ is the set of all possible pairs of fruits:

S = {A B, A M, O B, O M}

(iii) Probability of picking one apple and one banana (AB): First, let's find the probabilities of picking each fruit from its basket.

Basket A: $P (A) = \frac{1}{1 + 2} = \frac{1}{3}$ , $P (O) = \frac{2}{1 + 2} = \frac{2}{3}$
Basket B: $P (B) = \frac{1}{1 + 1} = \frac{1}{2}$ , $P (M) = \frac{1}{1 + 1} = \frac{1}{2}$

Now, calculate the probability of the specific outcome AB by multiplying the probabilities of the individual branches:

P (A B) = P (A from Basket A) \times P (B from Basket B)

P (A B) = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}

Answer: (i) The tree diagram is shown above. (ii) The sample space is ${A B, A M, O B, O M}$ . (iii) The probability of picking one apple and one banana is $\frac{1}{6}$ .

Remember
In a tree diagram for independent events, the probability of a particular sequence of outcomes is found by multiplying the probabilities of each event in the sequence. For mutually exclusive outcomes, you add their probabilities.

Exam-smart
You might be given a scenario with multiple steps (e.g., drawing balls from a bag twice, spinning a spinner and flipping a coin). You'll need to draw the tree diagram to list all possibilities or calculate the probability of a specific combined event. Sometimes, the probabilities of individual steps will be given, or you'll need to calculate them from provided counts.

The Mathematics of Maybe: Introduction to Probability — Notes

Read the notes

What is Probability?

What is Randomness?

Example of Randomness

The Probability Scale

Example Illustrating the Probability Scale

Example Problem on Probability Scale

Measuring Probability Objectively

Experimental Probability: Performing Observations or Experiments

Example

Example

Theoretical Probability

Example

Example

Theoretical Probability

Analysing Statistical Data Using Probability

Example 5: Favourite Fruit Survey

Elements of Probability: Sample Spaces and Events

Sample Space

Example 1: Rolling a Die and Tossing a Coin

Example 2: Choosing an Integer

Example 3: Balls in a Box

Example 4: Snack and Drink Combinations

Events

Examples of Events:

Example 5: Rolling a Die (Specific Event)

Example 6: Tossing Two Coins (Specific Event)

Tree Diagrams

Visualising Multi-Step Experiments

Example 1: Tossing a Fair Coin Twice

Calculating Probability of Combined Events

Example 2: Picking Fruits from Two Baskets

All Class 9 Mathematics chapters