1
A group of friends is playing a board game. The game involves rolling a standard six-sided die. One friend, Rohan, is concerned about the fairness of the die because he hasn't rolled a '6' in the last five rolls.(a) What is the theoretical probability of rolling a '6' on a fair six-sided die? [1 mark](b) Explain if Rohan's concern about the die being unfair is justified based on the concept of independent events and the Gambler's Fallacy. [2 marks](c) If Rohan rolls the die 100 times, how many times would he theoretically expect to roll a '6'? 1mark
2
A weather report states that there is a 75% chance of rain in your city tomorrow.(a) Represent this probability as a fraction and as a decimal. [1 mark](b) On a probability scale from 0 to 1, where would this event be placed and why? [1 mark](c) If the probability of rain is 0.75, what is the probability of it not raining? 1mark
3
A bag contains 5 red marbles and 3 blue marbles. A marble is drawn at random.(a) What is the total number of possible outcomes? [1 mark](b) What is the probability of drawing a red marble? [1 mark](c) What is the probability of drawing a blue marble? 1mark
4
Assertion: When tossing two fair coins simultaneously, the probability of getting two heads (HH) is 1/4.Reason: The possible outcomes when tossing two coins are HH, HT, TH, and TT. There are 4 equally likely outcomes, and only one of them is HH. 2marks
5
A student is creating a project on probability and decides to conduct an experiment by tossing a coin 50 times. They record the results as follows: 23 Heads and 27 Tails.(a) Calculate the experimental probability of getting heads. [1 mark](b) Calculate the experimental probability of getting tails. [1 mark](c) Compare the experimental probabilities with the theoretical probabilities for tossing a coin. Explain any observed differences. 2marks
6
A survey was conducted among 100 students to find their favourite subject. The results are as follows: Mathematics - 40, Science - 30, Social Studies - 20, English - 10.(a) What is the probability that a randomly selected student’s favourite subject is Mathematics? [1 mark](b) What is the probability that a randomly selected student’s favourite subject is not Science? 1mark
7
A teacher wants to illustrate the concept of sample space using a real-world example. She asks the students to consider the possible outcomes if a day's weather is classified as either sunny or rainy.(a) List the sample space for this experiment. [1 mark](b) If the teacher adds 'cloudy' as another possible outcome, what is the new sample space? 1mark
8
Consider the experiment of rolling a pair of dice.(a) What is the total number of possible outcomes? [1 mark](b) What is the probability of getting a sum of 7? 2marks
9
Assertion: In the word 'PROBABILITY', the probability of picking the letter 'A' is 1/11.Reason: There is one 'A' in the word 'PROBABILITY', which has 11 letters in total. 2marks
10
A fruit seller has a basket containing 6 apples, 4 oranges, and 5 bananas. If one fruit is picked at random, what is the probability that it is a banana? 2marks
11
A class of 30 students was surveyed about their preferred mode of transport to school. The results were: 15 students walk, 10 use bicycles, and 5 use public transport.(a) What is the probability that a randomly selected student walks to school? [1 mark](b) What is the probability that a randomly selected student uses public transport? [1 mark](c) If a student is chosen at random, estimate the number of students who use bicycles to get to school if the school has 600 students. 2marks
12
A spinner is divided into 8 equal sectors, numbered 1 to 8.(a) What is the probability of landing on an odd number? [1 mark](b) What is the probability of landing on a number greater than 5? 1mark
13
Two dice are rolled simultaneously.(a) List the sample space for the sum of the numbers obtained. [1 mark](b) What is the probability that the sum of the numbers is a prime number? 2marks
14
Assertion: When drawing a card from a standard deck of 52 playing cards, the probability of drawing a black card is 1/2.Reason: A standard deck has 26 black cards (13 spades and 13 clubs) and 26 red cards (13 hearts and 13 diamonds). 2marks
15
A company manufactures light bulbs. Out of 500 bulbs tested, 15 were found to be defective.(a) Calculate the experimental probability of a bulb being defective. [1 mark](b) Based on this, estimate the number of defective bulbs in a batch of 2000 bulbs. 2marks
16
A teacher explains that an event is a subset of a sample space.(a) Give an example of a sample space and an event related to it, based on tossing a single die. 2marks
17
A lottery has 100 tickets numbered from 1 to 100. 5 tickets are chosen as prizes. What is the probability that your ticket, numbered 42, is one of the prize-winning tickets? 2marks
18
A box contains 3 red balls, 2 blue balls, and 4 green balls.(a) What is the total number of balls in the box? [1 mark](b) What is the probability of drawing a green ball? [1 mark](c) What is the probability of drawing a ball that is neither red nor blue? 1mark
19
A child is asked to pick a day of the week at random.(a) What is the sample space for this experiment? [1 mark](b) What is the probability that the chosen day is a weekend day (Saturday or Sunday)? 1mark
20
A bag contains 10 red marbles, 8 blue marbles, and 7 green marbles. A marble is drawn at random.(a) What is the total number of marbles in the bag? [1 mark](b) What is the probability of drawing a blue marble? [1 mark](c) What is the probability of drawing a marble that is not green? 2marks
Answers
1
(a) The theoretical probability of rolling a '6' on a fair six-sided die is . 1mark(b) Rohan's concern is not justified. Each roll of a fair die is an independent event, meaning the outcome of previous rolls does not affect the outcome of future rolls. The Gambler's Fallacy is the mistaken belief that if an event has occurred many times, the opposite is more likely to occur. The probability of rolling a '6' remains for every roll, regardless of past results. [2 marks](c) He would theoretically expect to roll a '6' approximately times. So, about 17 times. [1 mark]
2
(a) As a fraction: . As a decimal: 0.75. 1mark(b) This event would be placed between 0.5 (equally likely) and 1 (certain), closer to 1, indicating that it is more likely than not to rain. [1 mark](c) The probability of it not raining is or . [1 mark]
3
(a) Total number of possible outcomes = Number of red marbles + Number of blue marbles = . 1mark(b) Probability of drawing a red marble = . [1 mark](c) Probability of drawing a blue marble = . [1 mark]
4
Assertion: True. Reason: True. The probability of getting two heads (HH) when tossing two fair coins is calculated as follows: The sample space is {HH, HT, TH, TT}. There are 4 equally likely outcomes. The favourable outcome is HH, which is 1. Therefore, P(HH) = . 2marks
5
(a) Experimental probability of getting heads = . 1mark(b) Experimental probability of getting tails = . [1 mark](c) The theoretical probability of getting heads is or , and the theoretical probability of getting tails is also or . The experimental probabilities ( for heads and for tails) are close to the theoretical probabilities but not exactly the same. This is because the experiment was conducted a finite number of times (50 tosses). As the number of trials increases, experimental probability tends to get closer to the theoretical probability (Law of Large Numbers). [2 marks]
6
(a) Probability (Favourite subject is Mathematics) = . 1mark(b) Probability (Favourite subject is not Science) = . [1 mark]
7
(a) The sample space S = {Sunny, Rainy}. 1mark(b) The new sample space S = {Sunny, Cloudy, Rainy}. [1 mark]
8
(a) When rolling a pair of dice, there are 6 possible outcomes for the first die and 6 for the second die. So, the total number of possible outcomes is . 1mark(b) The combinations that result in a sum of 7 are: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). There are 6 favourable outcomes.Probability of getting a sum of 7 = . [2 marks]
9
Assertion: True. Reason: True. The word 'PROBABILITY' has 11 letters: P, R, O, B, A, B, I, L, I, T, Y. The letter 'A' appears once. Thus, the probability of picking 'A' is . 2marks
10
Total number of fruits in the basket = .Probability of picking a banana = . 2marks
11
(a) Probability (Walks to school) = . 1mark(b) Probability (Uses public transport) = . [1 mark](c) From the sample, the probability of using bicycles is .Estimated number of students who use bicycles in the whole school = . [2 marks]
12
(a) The odd numbers are 1, 3, 5, 7. There are 4 odd numbers.Probability of landing on an odd number = . [1 mark](b) The numbers greater than 5 are 6, 7, 8. There are 3 such numbers.Probability of landing on a number greater than 5 = . 1mark
13
(a) The possible sums range from to . The sample space for the sum is {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. 1mark(b) The prime numbers in the sum range are 2, 3, 5, 7, 11.Sums of 2: (1,1) - 1 outcomeSums of 3: (1,2), (2,1) - 2 outcomesSums of 5: (1,4), (2,3), (3,2), (4,1) - 4 outcomesSums of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - 6 outcomesSums of 11: (5,6), (6,5) - 2 outcomesTotal favourable outcomes for a prime sum = .Probability of getting a prime sum = . [2 marks]
14
Assertion: True. Reason: True. A standard deck of 52 cards has 26 black cards (13 spades and 13 clubs) and 26 red cards (13 hearts and 13 diamonds). The probability of drawing a black card is . 2marks
15
(a) Experimental probability of a bulb being defective = . 1mark(b) Estimated number of defective bulbs in a batch of 2000 = . [2 marks]
16
(a) Sample Space (S) when tossing a single die: S = {1, 2, 3, 4, 5, 6}.Event (E) 'getting an even number': E = {2, 4, 6}. 2marks
17
Total number of tickets = 100.Number of prize-winning tickets = 5.The probability that ticket number 42 is one of the prize-winning tickets is the number of prize tickets divided by the total number of tickets.Probability = . 2marks
18
(a) Total number of balls in the box = . 1mark(b) Probability of drawing a green ball = . [1 mark](c) The balls that are neither red nor blue are the green balls.Probability of drawing a ball that is not green = Probability of drawing a red ball or a blue ball = . [1 mark]
19
(a) The sample space for picking a day of the week is S = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}. 1mark(b) There are 2 weekend days (Saturday and Sunday) out of 7 days.Probability of picking a weekend day = . [1 mark]
20
(a) Total number of marbles in the bag = . 1mark(b) Probability of drawing a blue marble = . [1 mark](c) The number of marbles that are not green = Number of red marbles + Number of blue marbles = .Probability of drawing a marble that is not green = . [2 marks]
