Probability | Exercise 14.1

Question 2

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) A player attempts to shoot a basketball. She/he shoots or misses the shot.

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

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

Solution

Understand the Question

Outcomes of an experiment are equally likely if each outcome has the exact same chance of occurring.
If external factors, conditions, or skill cause one outcome to be more probable than another, the outcomes are not equally likely.
Having only two outcomes (a binary choice) does not automatically mean each has a $50\%$ chance.

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

Step 1 · Analyze Outcomes of Starting a Car

The possible outcomes are that the car starts or does not start. Whether the car starts depends on external factors such as the mechanical condition of the car, fuel level, and battery status.

Since these factors influence the result, the two outcomes do not have an equal chance of occurring.

Answer

(i) Not equally likely

(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.

Step 1 · Analyze Outcomes of Shooting a Basketball

The possible outcomes are that the player makes the shot or misses the shot. The probability of scoring depends on the player's skill, training, defense, and position.

Therefore, the two outcomes do not have an equal chance of occurring.

Answer

(ii) Not equally likely

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

Step 1 · Analyze Outcomes of Answering a True-False Question

A true-false question has only two possible options, and exactly one of them is correct.

When guessing, each outcome (right or wrong) has an equal probability: $P(\text{Right}) = P(\text{Wrong}) = \dfrac{1}{2}$

Therefore, the outcomes are equally likely.

Answer

(iii) Equally likely

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

Step 1 · Analyze Outcomes of a Baby's Gender

There are only two mutually exclusive outcomes: the baby is a boy or a girl. Biologically, each outcome has an equal probability in a random birth: $P(\text{Boy}) = P(\text{Girl}) = \dfrac{1}{2}$

Therefore, the outcomes are equally likely.

Answer

(iv) Equally likely

Common Mistakes

Binary Outcomes Fallacy: Assuming that any scenario with only two possible outcomes (like starting a car) must automatically be equally likely with a $50\%$ probability.
Ignoring Contextual Variables: Overlooking real-world variables such as player skill, preparation, or machine health when determining theoretical probability.

More questions in Exercise 14.1

Complete the following statements:

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) A player attempts to shoot a basketball. She/he shoots or misses the shot.

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

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

Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?

Which of the following cannot be the probability of an event?

(A) $\dfrac{2}{3}$

(B) $-1.5$

(D) $0.7$

If $P(E) = 0.05$ , what is the probability of 'not E'?

A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out

(i) an orange flavoured candy?

(ii) a lemon flavoured candy?

It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is

(i) red ?

(ii) not red?

A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be

(i) red?

(ii) white?

(iii) not green?

Q10

A piggy bank contains hundred 50p coins, fifty ₹ 1 coins, twenty ₹ 2 coins and ten ₹ 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin

(i) will be a 50 p coin ?

(ii) will not be a ₹ 5 coin?

Q11

Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish (see Fig. 14.4). What is the probability that the fish taken out is a male fish?

Q12

A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 (see Fig. 14.5), 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?

Q13

A die is thrown once. Find the probability of getting

(i) a prime number;

(ii) a number lying between 2 and 6;

(iii) an odd number.

Q14

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting

(i) a king of red colour

(ii) a face card

(iii) a red face card

(iv) the jack of hearts

(v) a spade

(vi) the queen of diamonds

Q15

Five cards—the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random.

(i) What is the probability that the card is the queen?

(ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?

Q16

12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.

Q17

(i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective? (ii) Suppose the bulb drawn in (i) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective?

Q18

A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears

(i) a two-digit number

(ii) a perfect square number

(iii) a number divisible by 5.

Q19

A child has a die whose six faces show the letters as given below:

The die is thrown once. What is the probability of getting (i) A? (ii) D?

Q20

Suppose you drop a die at random on the rectangular region shown in Fig. 14.6. What is the probability that it will land inside the circle with diameter $1 \text{ m}$ ?

Q21

A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that

(i) She will buy it ?

(ii) She will not buy it ?

Q22

Refer to Example 13.

(i) Complete the following table:

(ii) A student argues that 'there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability $\dfrac{1}{11}$ . Do you agree with this argument? Justify your answer.

Q23

A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

Q24

A die is thrown twice. What is the probability that

(i) 5 will not come up either time?

(ii) 5 will come up at least once?

[Hint : Throwing a die twice and throwing two dice simultaneously are treated as the same experiment]

Q25

Which of the following arguments are correct and which are not correct? Give reasons for your answer.

(i) If two coins are tossed simultaneously there are three possible outcomes—two heads, two tails or one of each. Therefore, for each of these outcomes, the probability is $\dfrac{1}{3}$ .

(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 $\dfrac{1}{2}$ .