Practice Questions

A single die is thrown. Calculate the probability of getting a composite number.

Evaluate whether the following can represent the probabilities of all elementary events of an experiment: $P(E_1) = 0.2$, $P(E_2) = 0.3$, $P(E_3) = 0.4$, and $P(E_4) = 0.2$. Justify your answer.

What is the maximum value for the probability of any event?

State the formula for the theoretical probability of an event E, assuming equally likely outcomes.

If the probability of winning a game is 0.38, calculate the probability of losing the game.

Define an elementary event.

A letter is chosen at random from the word 'MATHEMATICS'. Calculate the probability that the chosen letter is a vowel.

Identify the probability of an impossible event.

Two unbiased coins are tossed simultaneously. Calculate the probability of getting (i) at most one head, (ii) exactly two tails.

Describe the contents of a standard deck of 52 playing cards.

Explain the concept of 'equally likely outcomes' using the example of throwing a fair die.

A student claims that since a standard die has six faces, and three are odd (1, 3, 5) and three are even (2, 4, 6), the event of rolling an odd number and the event of rolling an even number are complementary. Justify this claim.

A box contains cards numbered from 1 to 20. A card is drawn at random. Formulate events E (drawing a prime number) and F (drawing a multiple of 4). Justify whether these two events are mutually exclusive.

Design a simple game using a standard deck of 52 playing cards where the probability of winning is exactly $\frac{3}{13}$. Formulate the rules for winning and justify that the probability is correct.

Formulate a real-world scenario, not involving games of chance (like dice or cards), where the outcomes are not equally likely. Justify your reasoning.

A piggy bank contains one hundred 50p coins, seventy Rs. 1 coins, fifty Rs. 2 coins and thirty Rs. 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, calculate the probability that the coin will be (i) a Rs. 1 coin, (ii) not a Rs. 5 coin, (iii) will have a value less than Rs. 2.

Name the event which is the complement of event E and provide its notation.

Describe the relationship between the probability of an event and the probability of its complementary event.

A box contains 6 green marbles and 10 yellow marbles. Explain why the outcomes 'drawing a green marble' and 'drawing a yellow marble' are not equally likely.

List all the possible outcomes when a pair of dice is thrown once. State the total number of possible outcomes.

If you toss a coin 100 times and get 45 heads, what is the empirical probability of getting a head? How does this differ from the theoretical probability?

Summarize the key assumptions made when calculating theoretical probability. Use the example of drawing a ball from a bag containing 4 red and 1 blue ball to illustrate a situation where not all outcomes are equally likely.

From a well-shuffled deck of 52 cards, one card is drawn at random. Calculate the probability that the card is a black-coloured face card.

A bag contains 15 balls, of which $x$ are red. If a ball is drawn at random, the probability of drawing a red ball is $\frac{1}{3}$. Calculate the value of $x$.

A box contains discs numbered from 1 to 50. A disc is drawn at random. Analyze the outcomes and calculate the probability that the number on the disc is (i) a perfect square, (ii) divisible by both 2 and 3.

A bag contains 6 red, 4 blue, and 8 green balls. One ball is drawn at random. Calculate the probability that the ball drawn is (i) a blue ball, (ii) not a red ball, (iii) red or green.

Two different dice are rolled together. Analyze the possible sums and calculate the probability that the sum of the numbers on the two dice is a multiple of 3.

A game of chance has a spinner with four sectors coloured red, blue, green, and yellow. The probabilities of the pointer landing on red, blue, and green are 0.25, 0.40, and 0.15 respectively. Calculate the probability of the pointer landing on yellow.

Two different dice, one red and one blue, are thrown simultaneously. Analyze all possible outcomes and calculate the probability of the following events: (i) The product of the numbers on the top of the dice is 6. (ii) The number on the red die is greater than the number on the blue die.

Critique the following statement: 'If a family has two children, and we know one of them is a boy, the probability that the other child is also a boy is $\frac{1}{2}$'. Justify your reasoning by listing the sample space.

A friend argues that when two dice are rolled, there are 11 possible sums (2 through 12), so the probability of getting any specific sum, for example 7, is $\frac{1}{11}$. Critique this argument and justify your conclusion with correct calculations.

Two friends are playing a game with a fair six-sided die. Player A wins if the outcome is a factor of 6. Player B wins if the outcome is a multiple of 3. Evaluate if this game is fair. Justify your conclusion.

A bag contains 24 balls, of which $x$ are red, $2x$ are white, and $3x$ are blue. A ball is drawn at random. Create a model to find the number of white balls in the bag and justify your steps.

It is known that a box of 100 computer chips contains 5 defective ones. You are designing a quality control test where two chips are selected at random, one after another, without replacement. Formulate the probability that both selected chips are defective. Justify each part of your calculation.

Justify why the theoretical probability of an event is always a rational number, based on its definition.

Formulate an expression for the probability that a randomly selected year in the 21st century (from 2001 to 2100, inclusive) is a leap year. Justify the numerator and denominator used in your expression.

A student makes two arguments about probability. Identify if each argument is correct or not and explain your reasoning. (i) "If two coins are tossed, there are three possible outcomes: two heads, two tails, or one of each. Therefore, the probability of each outcome is $\frac{1}{3}$." (ii) "If a die is thrown, there are two possible outcomes: an odd number or an even number. Therefore, the probability of getting an odd number is $\frac{1}{2}$."

All the aces and kings are removed from a standard deck of 52 playing cards. A card is then drawn at random from the remaining deck. Calculate the probability of getting (i) a red card, (ii) a face card.

A game is designed as follows: A player tosses three fair coins. The player wins if the number of heads is a prime number, and loses otherwise. Critique the fairness of this game. Justify your conclusion by calculating the probability of winning and losing.

An experiment consists of tossing three coins simultaneously. (i) List all the possible outcomes for this experiment. (ii) List the outcomes for the event 'getting exactly two heads'. (iii) List the outcomes for the event 'getting at least one tail'. (iv) List the outcomes for the event 'getting no heads'. (v) Explain if the event 'getting no heads' is an elementary event.

A carton of 150 light bulbs contains 18 defective bulbs. A bulb is drawn at random from the carton. Calculate the probability that the bulb drawn is not defective. If the bulb drawn is not defective and not replaced, and another bulb is drawn, what is the new probability of drawing a non-defective bulb?

Design a fair game for two players, A and B, using two standard six-sided dice. Formulate the rules for winning for each player, based on the product of the numbers shown on the dice. Justify the fairness of your game by calculating the probabilities of winning for each player.

Explain why the probability of an event must be a value between 0 and 1, inclusive.

A box contains 90 discs numbered 1 to 90. One disc is drawn at random. Analyze the numbers and calculate the probability that the disc bears: (i) a two-digit number, (ii) a number which is a perfect cube, (iii) a prime number less than 20.