Question

A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected.

Find P(A and B) and find whether the events A and B are independent events or not.

Answer 1

Since two cards are picked with replacement from 6 cards, the total number of outcomes is:

n(S) = 6 × 6

= 36

The possible outcomes for a sum of 10 are:

A = {(4, 6). (5, 5), (6, 4)}

n(A) = 3

p(A) = `3/36`

= `1/12`

First card can be 1, 2, 3, 5, or 6

n(B) = 5 × 6 = 30

P(B) = `30/36`

= `5/6`

P(A ∩ B)

A ∩ B refers to the outcomes in \(A\) where the first card is not 4.

Looking at event A:

(4, 6) - First card is 4 (Exclude)

(5, 5) - First card is not 4 (Include)

(6, 4) - First card is not 4 (Include)

so,

A ∩ B = {(5, 5), (6, 4)}

n(A ∩ B) = 2

P(A ∩ B) = `2/36`

= `1/18`

Two events are independent if P(A ∩ B) = P(A) . P(B)

P(A) . P(B) = `1/12 xx 5/6`

= `5/72`

P(A ∩ B) = `1/18`

= `4/72`

Since, `4/72 ≠ 5/72`, the events A and B are not independent.