The Mathematics of Maybe: Introduction to Probability | EOT

Question 3

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.

Solution

Understand the Question

Equally Likely Outcomes: Outcomes of an experiment are said to be equally likely if each possible outcome has the exact same probability of occurring.
If the outcomes depend on external factors (such as the mechanical condition of an object) or if the distribution is uneven, the outcomes are not equally likely.

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

Step 1 · Analyze the Outcomes

Whether a car starts or does not start depends on various mechanical factors such as fuel level, battery condition, and engine health. A well-maintained car usually starts, whereas a damaged one may not. Hence, the two outcomes do not have an equal chance of occurring.

Answer

(i) Not equally likely

(ii) Tossing a fair coin once.

Step 1 · Analyze the Outcomes

When a fair coin is tossed, there are two possible outcomes: Head and Tail.

$P(\text{Head}) = P(\text{Tail}) = \dfrac{1}{2}$

Both outcomes have the same probability.

Answer

(ii) Equally likely

(iii) Rolling a fair 6-sided die.

Step 1 · Analyze the Outcomes

A fair six-sided die has six numbered faces from $1$ to $6$ . Each face has an equal probability of showing on top.

$P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = \dfrac{1}{6}$

Answer

(iii) Equally likely

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

Step 1 · Analyze the Outcomes

Total number of marbles $= 3 + 7 = 10$ .

$P(\text{Red}) = \dfrac{3}{10}$

$P(\text{Blue}) = \dfrac{7}{10}$

Since $P(\text{Red}) \neq P(\text{Blue})$ , drawing a red marble and drawing a blue marble do not have equal probabilities.

Answer

(iv) Not equally likely

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

Step 1 · Analyze the Outcomes

Under standard biological and theoretical probability assumptions, the birth of a boy or a girl is mutually exclusive and equally probable.

$P(\text{Boy}) = P(\text{Girl}) = \dfrac{1}{2}$

Therefore, the outcomes are equally likely.

Answer

(v) Equally likely

Common Mistakes

Binary Trap: Assuming that having only two possible outcomes (e.g. starting vs not starting, red marble vs blue marble) automatically makes them equally likely with $50\%$ probability each.
Ignoring Composition: Overlooking the quantity of items when calculating probabilities (e.g. 3 red vs 7 blue marbles).

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 $1\text{ m}$ ?