The Mathematics of Maybe: Introduction to Probability | EOT

Question 4

Write the sample space and calculate the probability based on the given information.

(i) Two coins are tossed at the same time. What is the probability of getting at least one head?

(ii) Ten identical cards numbered 1 to 10 are placed in a box. One card is drawn at random. What is the probability of drawing a card with an even number?

(iii) A die is rolled once. What is the probability of getting a number greater than 4?

(iv) A bag contains 3 red balls, 2 blue balls, and 1 green ball. One ball is picked at random. What is the probability that it is not red?

(v) Three coins are tossed simultaneously. What is the probability of getting exactly two heads?

Solution

Understand the Question

The sample space ( $S$ ) is the set of all possible outcomes of a random experiment, and $n(S)$ is the total number of outcomes.
The probability of an event $E$ is calculated using the formula:

$P(E) = \dfrac{\text{Number of favourable outcomes } n(E)}{\text{Total number of possible outcomes } n(S)}$

To solve each part, list the sample space $S$ , determine the favourable outcomes for event $E$ , and evaluate $P(E)$ .

(i) Two coins are tossed at the same time. What is the probability of getting at least one head?

Step 1 · List Sample Space and Favourable Outcomes

When two coins are tossed simultaneously, the sample space is: $S = \{\text{HH}, \text{HT}, \text{TH}, \text{TT}\}$ $n(S) = 4$

Let $E$ be the event of getting at least one head (1 or 2 heads): $E = \{\text{HH}, \text{HT}, \text{TH}\}$ $n(E) = 3$

Step 2 · Calculate Probability

\begin{aligned} P(E) &= \dfrac{n(E)}{n(S)} \\[0.6em] &= \dfrac{3}{4} \end{aligned}

Answer

(i) $\dfrac{3}{4}$

(ii) Ten identical cards numbered 1 to 10 are placed in a box. One card is drawn at random. What is the probability of drawing a card with an even number?

Step 1 · List Sample Space and Favourable Outcomes

The sample space of cards numbered 1 to 10 is: $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ $n(S) = 10$

Let $E$ be the event of drawing an even number: $E = \{2, 4, 6, 8, 10\}$ $n(E) = 5$

Step 2 · Calculate Probability

\begin{aligned} P(E) &= \dfrac{n(E)}{n(S)} \\[0.6em] &= \dfrac{5}{10} \\[0.6em] &= \dfrac{1}{2} \end{aligned}

Answer

(ii) $\dfrac{1}{2}$

(iii) A die is rolled once. What is the probability of getting a number greater than 4?

Step 1 · List Sample Space and Favourable Outcomes

When a die is rolled, the sample space is: $S = \{1, 2, 3, 4, 5, 6\}$ $n(S) = 6$

Let $E$ be the event of getting a number strictly greater than 4: $E = \{5, 6\}$ $n(E) = 2$

Step 2 · Calculate Probability

\begin{aligned} P(E) &= \dfrac{n(E)}{n(S)} \\[0.6em] &= \dfrac{2}{6} \\[0.6em] &= \dfrac{1}{3} \end{aligned}

Answer

(iii) $\dfrac{1}{3}$

(iv) A bag contains 3 red balls, 2 blue balls, and 1 green ball. One ball is picked at random. What is the probability that it is not red?

Step 1 · Find Total and Favourable Outcomes

Total number of balls in the bag: $n(S) = 3 + 2 + 1 = 6$

Let $E$ be the event of picking a ball that is not red (i.e., blue or green): $n(E) = 2 + 1 = 3$

Step 2 · Calculate Probability

\begin{aligned} P(E) &= \dfrac{n(E)}{n(S)} \\[0.6em] &= \dfrac{3}{6} \\[0.6em] &= \dfrac{1}{2} \end{aligned}

Answer

(iv) $\dfrac{1}{2}$

(v) Three coins are tossed simultaneously. What is the probability of getting exactly two heads?

Step 1 · List Sample Space and Favourable Outcomes

When three coins are tossed simultaneously, the total number of outcomes is $2^3 = 8$ : $S = \{\text{HHH}, \text{HHT}, \text{HTH}, \text{THH}, \text{HTT}, \text{THT}, \text{TTH}, \text{TTT}\}$ $n(S) = 8$

Let $E$ be the event of getting exactly two heads: $E = \{\text{HHT}, \text{HTH}, \text{THH}\}$ $n(E) = 3$

Step 2 · Calculate Probability

\begin{aligned} P(E) &= \dfrac{n(E)}{n(S)} \\[0.6em] &= \dfrac{3}{8} \end{aligned}

Answer

(v) $\dfrac{3}{8}$

Common Mistakes

"At least" vs "Exactly": "At least one head" means $\ge 1$ heads (includes 1 and 2 heads), while "exactly two heads" allows only outcomes with precisely 2 heads.
Strict Inequalities: "Greater than 4" does not include the number 4 itself; only 5 and 6 are favourable outcomes.
Simplified Form: Always simplify fractions to their lowest terms (e.g., $\dfrac{5}{10} = \dfrac{1}{2}$ , $\dfrac{2}{6} = \dfrac{1}{3}$ ).

More questions in EOT

Fill in the blanks.

(i) The probability of an impossible event is _______.

(ii) The set of all possible outcomes of a random experiment is called the _______.

(iii) The probability of an event that is certain to happen is _______.

(iv) Tossing a fair coin has a probability of _______ for getting heads.

Which of the following experiments have equally likely outcomes? Explain.

(i) A driver attempts to start a car. The car starts or does not start.

(ii) Tossing a fair coin once.

(iii) Rolling a fair 6-sided die.

(iv) Choosing a marble randomly from a bag that contains 3 red marbles and 7 blue marbles.

(v) A baby is born. It is a boy or a girl.

Write the sample space and calculate the probability based on the given information.

(i) Two coins are tossed at the same time. What is the probability of getting at least one head?

(ii) Ten identical cards numbered 1 to 10 are placed in a box. One card is drawn at random. What is the probability of drawing a card with an even number?

(iii) A die is rolled once. What is the probability of getting a number greater than 4?

(iv) A bag contains 3 red balls, 2 blue balls, and 1 green ball. One ball is picked at random. What is the probability that it is not red?

(v) Three coins are tossed simultaneously. What is the probability of getting exactly two heads?

A bag has 3 candies: strawberry, lemon, and mint. One is picked at random. What is the probability of picking a strawberry candy?

A child has 2 shirts (one red and one blue) and 3 types of pants (jeans, khakis, and shorts). List all the possible combinations of outfits consisting of one shirt and one pair of pants. Display your answer in a table format.

A tyre company records distances before replacement in 1000 cases.

Find the probability that a randomly chosen tyre lasts:

(i) Less than $4000\text{ km}$ .

(ii) Between $4000$ and $14000\text{ km}$ .

(iii) More than $14000\text{ km}$ .

The letters of the word 'PEACE' are placed on cards. Leela draws a card without looking.

(i) What is the probability that it is a P, E or C?

(ii) What is the probability that it is not an E?

A game of chance consists of spinning an arrow (see Fig. 7.7.) which comes to rest pointing at one of the numbers $1, 2, 3, 4, 5, 6, 7, 8$ , and these are equally likely outcomes. What is the probability that it will point at

(i) $8$ ?

(ii) An odd number?

(iii) A number greater than $2$ ?

(iv) A number less than $9$ ?

(v) A multiple of $3$ ?

Q10

A basket contains 4 red balls and 5 blue balls. One ball is drawn and laid aside, and a second ball is drawn. Draw a tree diagram to represent the possible outcomes and probabilities. Use the tree diagram to answer the following questions.

(i) What is the probability of drawing a red ball and then a blue ball?

(ii) What is the probability of drawing 2 blue balls?

Q11

I throw a pair of 6-sided dice. Write down an event that has a probability of $0$ and an outcome that has a probability of $1$ .

Q12

Write the sample space and calculate the probability based on the given information.

(i) Two dice are rolled. What is the probability that the sum is a prime number greater than 5?

(ii) A bag contains 4 red, 3 green, and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are of different colours?

(iii) Three coins are tossed. What is the probability that the first coin shows heads and exactly two heads occur in total?

(iv) A four-digit number is formed using the digits 1, 2, 3, and 4 with no repetition. What is the probability that the number is even?

(v) A student takes a multiple-choice test with 3 questions, each having 4 options (A, B, C, D), with only one correct answer. What is the probability that the student guesses and gets exactly 2 answers correct?

Q13

A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram for the following experiments:

(i) A ball is drawn, and the number is recorded. Then the ball is returned, and a second ball is drawn and recorded.

(ii) A ball is drawn and recorded. Without replacing the first ball, the experimenter draws and records a second ball.

(iii) What are the sizes of these two sample spaces?

Q14

List the elements of a sample space for the simultaneous tossing of a coin and drawing of a card from a set of 6 cards numbered 1 through 6.

Q15

Three coins are tossed, and the number of heads is recorded. Which of the following lists is a sample space for this experiment? Why do the other lists fail to qualify as a sample space?

(i) $\{1, 2, 3\}$

(ii) $\{0, 1, 2\}$

(iii) $\{0, 1, 2, 3, 4\}$

(iv) $\{0, 1, 2, 3\}$

Q16

Suppose you drop a dye at random on the rectangular region shown in Fig. 7.8. What is the probability that it will land inside the circle with a diameter of $1\text{ m}$ ?