Exercise 2A - Probability of Combined Events

Exercise 2b

Overview

Exercise 2A from New Syllabus Mathematics Book 4 (8th Edition) focuses on the topic Probability of Combined Events. This exercise includes 26 real-world and theoretical probability problems that test understanding of sample spaces, event probabilities, and compound event reasoning.

Through engaging scenarios using dice, spinners, cards, marbles, and surveys, students explore mathematical reasoning supported by set notation, Venn diagrams, and tree diagrams.

List of Questions in Exercise 2A

List the sample space of each of the following probability experiments:
1. A fair 12-sided die, numbered from 1 to 12, is rolled.
2. A fair coin is tossed.
3. A ball is drawn randomly from a bag containing 4 black balls and 2 white balls, where the balls are identical except for their colour.
4. A letter is chosen at random from the word "STUDENTS".
There are 2 green marbles and 3 yellow marbles in a bag. The marbles are identical except for their colour. A marble is drawn at random.
1. List the sample space in set notation.
2. Find the probability that the marble drawn is (a) yellow, (b) not yellow.
3. What is the probability the marble drawn is (a) black, (b) green or yellow?
Each letter of the word "PROBABILITY" is written on identical cards. One is chosen at random.
1. List the sample space in set notation.
2. Find the probability the chosen letter is (a) B, (b) T, (c) I or O, (d) E.
3. Find the probability the chosen letter is (a) a vowel, (b) a consonant.
All eight pangolin species are protected under international laws. In Singapore, the method of capture-recapture (as described in the Chapter Opener) is used to monitor the pangolin population in the nature reserves. In a particular year, 22 injured pangolins were rescued and tagged. These pangolins were then released back into the nature reserves after they recovered. Over period of one year, five pangolins were captured on camera, of which only one was tagged. Estimate the number of pangolins that lived in the nature reserves. State the assumptions you have made to solve this problem.
A survey was conducted to find out which of the four activities, soccer, basketball, chess and reading, students in a class prefer. The pie chart shows the results of the survey. A Student is selected at random. Find the probability that the student prefers.
1. soccer.
2. reading.
3. basketball or chess.
A spinner consists of a board in the shape of a regular octagon with centre O and a spinning pointer. The board is divided into 4 regions and the spinning pointer is spun at random. Find the probability that the spinning pointer will stop in
1. region S.
2. region Q.
3. region P or R.
A box contains three cards bearing the numbers 1,2 and 3. A second box contains four cards bearing the numbers 2,3,4 and 5. A card is chosen at random from each box.
1. Display all the possible outcomes of the experiment using a sample space diagram.
2. With the help of the sample space diagram, calculate the probability that
  (a) the cards bear the same number,
  (b) the numbers on the cards are different,
  (c) both numbers are prime numbers,
  (d) exactly one number is a multiple of 2.
Bag P contains a red, a blue and a white marble while bag Q contains a blue and a red marble. The marble are identical except for their colour. A marble is picked at random from each bag.
1. Display all the possible outcomes of the experiment using a sample space diagram.
2. With the help of the sample space diagram, find the probability that the two marbles selected are
  (a) of the same colour,
  (b) blue and red,
  (c) of different colours.
Six cards numbered 0,1,2,3,4 and 5 are placed in a box. A card is drawn at random from the box and the number on the card is noted before it is replaced in the box. A second card is then drawn at random from the box and the sum of the two numbers is obtained. The sample space diagram below shows some of the values of the sum of the two numbers.
1. Copy and complete the sample space diagram.
2. How many possible outcomes are there in the sample space of this experiment?
3. What is the probability that the sum of the two numbers is:
  (a) 7,
  (b) a prime number,
  (c) not a prime number,
  (d) even,
  (e) not even?
4. Which of the two sums of the two numbers is more likely to occur, 7 or 8? Explain.
It is given that X = {4, 5, 6}, and Y = {7, 8, 9}. An element x is selected at random from X and an element y is selected at random from Y. The sample space diagrams display separately some of the values of x+y and xy.
1. Copy and complete the sample space diagrams.
2. Find the probability that the sum x+y is
  (a) prime,
  (b) greater than 12,
  (c) at most 14.
3. Find the probability that the product xy is:
  (a) odd,
  (b) even,
  (c) at most 40.
A fair coin is tossed three times.
1. Display all possible outcomes of the experiment using a tree diagram.
2. With the help of the tree diagram, find the probability of obtaining
  (a) three tails,
  (b) exactly two tails,
  (c) at least two tails,
  (d) no tails.
Three ladies are happily awaiting the arrival of their bundles of joy within the year. Display the sample space of the genders of the three babies using an appropriate diagram, assuming that the babies are equally likely to be either a boy or a girl. Hence, find the probability that there will be a total of
1. three baby boys,
2. two baby boys and one baby girl
3. one baby boy and two baby girls.
A spinner, with three equal sectors and a spinning pointer as shows in the diagram, and a fair coin are used in a game. The pointer is spun once and the coin is tossed once. Each time the pointer is spun, it is equally likely to stop at any sector.
1. Display all possible outcomes of the experiment using a tree diagram.
2. With the help of the tree diagram, calculate the probability of getting
  (a) red on the spinner and tail on the coin,
  (b) blue or orange on the spinner and head on the coin.
The Venn diagram shows the number of teenagers who play cricket and the number of teenagers who play soccer in a neighbourhood, where C is the set of teenagers who play cricket and S is the set of teenagers who play soccer.
A teenager is chosen at random. What is the probability that this teenager
1. plays either cricket or soccer,
2. only plays cricket,
3. plays cricket,
4. does not play either sport.
A card is drawn at random from a standard pack of 52 playing cards. Find the probability of drawing
1. the Queen of diamonds,
2. a black card,
3. a spade,
4. a card which is not a spade.
A two-digit number is randomly formed using the digits 1,4 and 6. Repetition of digits is allowed.
1. List the sample space in set notation.
2. By using set notation for the respective events, find the probability that the two-digit number formed is
  (a) divisible by 3,
  (b) a perfect square,
  (c) a prime number,
  (d) a composite number.
ABCDEF is a regular hexagon with centre 0. M is the midpoint of A B and N is the midpoint of BC. A point is selected at random in the regular hexagon. Find the probability that the point lies in the kite MBNO.
In an experiment, two spinners are constructed with spinning pointers as shown in the diagrams below. Both pointers are spun. Each time the pointer is spun, it is equally likely to stop at any sector.
1. Find the probability that the pointers will point at
  (a) numbers on the spinners whose sum is 6,
  (b) the same number on both spinners,
  (c) different numbers on the spinners,
  (d) two different prime numbers.
2. What is the porbability that the number on the first spinner will be less than the number on the second spinner?
3. What is the probability that the larger of the two numbers on the spinners is 3?
In a game, the player tosses a fair coin and rolls a fair 6-sided die simultaneously. If the coin shows a head, the player's score is the score on the die. If the coin shows a tail, then the player's score is twice the score on the die. Some of the player's possible scores are shown in the sample space diagram.
1. Copy and complete the sample space diagram.
2. Using the sample space diagram, find the probability that the player's score is
  (a) odd,
  (b) even,
  (c) a prime number,
  (d) less than or equal to 8,
  (e) a multiple of 3.
Two fair 6-sided dice were rolled together and the difference between the numbers on their faces was calculated. Some of the differences are shown in the sample space diagram below.
1. Copy and complete the sample space diagram.
2. Using the sample space diagram, find the probability that the difference between the two numbers is
  (a) 1,
  (b) non-zero,
  (c) odd,
  (d) a prime number,
  (e) greater than 2.
A bag contains 3 cards numbered 1, 3 and 5. A second bag contains 3 cards numbered 1, 2 and 7. One card is drawn at random from each bag.
1. Display all the possible outcomes of the experiment using a tree diagram.
2. With the help of the tree diagram, calculate the probability that the two numbers obtained
  (a) are both odd,
  (b) are both prime,
  (c) have a sum greater than 4,
  (d) have a sum that is even,
  (e) have a product that is prime,
  (f) have a product that is greater than 20,
  (g) have a product that is divisible by 7.
The elements of ξ of P and of Q are shown in the Venn diagram below. An element is selected at random. Find the probability that the selected element is
1. an even number,
2. a number not in P or Q,
3. a prime number,
4. a prime number in Q,
A fair coin and a fair 6-sided die are tosssed and rolled respectively. Using set notation, list the sample space of the experiment.
Hotel Y is a two-storey hotel, with rooms (and their respective room numbers) arranged as shown in the diagram below. Rooms are allocated at random when guests arrive and each guest is allocated one room. Cheryl and Joyce arrive at Hotel Y on a particular day. Upon their arrival, none of the rooms in Hotel Y are occupied.
1. With the help of a sample space diagram, find the probability taht Cheryl and Joyce.
  (a) stay next to each other,
  (b) stay on different storeys,
  (c) do not stay next to each other,
2. Suppose the hotel accepts Cheryl's request that she only wants to be allocated rooms on the second floor, what is the probability of her staying next to Joyce?
Two fair tetrahedral dice and a fair 6-sided die are rolled simultaneously. The numbers on the tetrahedral dice are 1, 2, 3 and 4 while the numbers on the 6-sided die are 1, 2, 3, 4, 5 and 6. What is the probability that the score on the 6-sided die is greater than the sum of the scores on the two tetrahedral dice?
The Venn diagram shows the number of elements of ξ, of A, of B and of C. It is given that n(ξ) = 48 and n(A ∩ B) = n(C′).
1. Find the values of x and y.
2. An element is selected at random. Find
  (a) P(C),
  (b) P(A ∪ B).

You can view Exercise 2A online or download the complete section in PDF. Step-by-step solutions will be added for each question to support your learning journey.