Probability — Assertion-Reason Type Questions

Assertion (A): A bag contains 4 red, and 8 blue marbles. If a marble is drawn at random, then the probability of drawing a red marble is $\Big(\dfrac{1}{3}\Big)$.

Reason (R): Probability of an event A is given by P(A) = $\dfrac{\text{total no. of possible outcomes}}{\text{no. of favourable outcomes}}$.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Total number of outcomes = 4 + 8 = 12

Let E be the event of drawing a red marble.

The number of favorable outcomes of the event E = 4.

$\therefore P(E) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \dfrac{4}{12} = \dfrac{1}{3}$

Therefore, Assertion (A) is true.

The formula given in Reason (R) is incorrect. The correct formula for probability is the reciprocal of what is stated:

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

Therefore, Reason (R) is false.

A is true, but R is false.

Hence, option 3 is the correct option.

Assertion (A): The probability of not picking a face card when you draw a card at random from a deck of playing cards is $\Big(\dfrac{3}{13}\Big)$.

Reason (R): There are 12 face cards in a deck of playing cards.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Total number of cards = 52

Number of face cards = 12

Let E be the event of not picking a face card.

The number of favorable outcomes of the event E = 52 - 12 = 40

$\therefore P(E) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \dfrac{40}{52} = \dfrac{10}{13}$

The probability of not picking a face card is $\dfrac{10}{13}$, not $\dfrac{3}{13}$.

Therefore, Assertion (A) is false.

In a standard deck of cards, the face cards are the King, Queen, and Jack.

Each of the 4 suits (Hearts, Diamonds, Clubs, Spades) has 3 face cards.

Total face cards = 4 × 3 = 12.

Therefore, Reason (R) is true.

A is false, but R is true.

Hence, option 4 is the correct option.

Assertion (A): A die is thrown twice. The probability of getting a bigger value on the first throw is $\Big(\dfrac{5}{12}\Big)$.

Reason (R): When we throw a die once, total number of possible outcomes is 6 and when the die is thrown twice, the total number of possible outcomes is 6 + 6 = 12.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

When a die is thrown twice,

Total number of outcomes = 6 × 6 = 36

Let E be the event of getting a bigger value on the first throw,

E = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}

The number of favorable outcomes of the event E = 15

$\therefore P(E) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \dfrac{15}{36} = \dfrac{5}{12}$

Therefore, Assertion (A) is true.

When a die is thrown twice, the total number of outcomes is the product of the outcomes of each throw:

Total Outcomes = 6 × 6 = 36, not 6 + 6 = 12.

Therefore, Reason (R) is false.

A is true, but R is false.

Hence, option 3 is the correct option.

Assertion (A): A die is thrown once and the probability of getting an even number is $\dfrac{2}{3}$.

Reason (R): The sample space for even numbers on a die is {2, 4, 6}

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Sample space when a die is thrown = {1, 2, 3, 4, 5, 6}.

Sample space for even numbers = {2, 4, 6}

Probability of getting an even number = $\dfrac{3}{6} = \dfrac{1}{2}$.

∴ A is false and R is true.

Hence, Option 4 is the correct option.