Question

Two dice are thrown together. Let A be the event ‘getting 6 on the first die’ and B be the event ‘getting 2 on the second die’. Are the events A and B independent?

Answer 1

Answer:

A and B are independent.

Step-by-step explanation:

Two dice are thrown together. Let A be the event ‘getting 6 on the first die’ and B be the event ‘getting 2 on the second die’. Are the events A and B independent?

To find events A and B are independent, satisfy the given condition

$\boxed{P(A).P(B) = P(A \cap \: B)}$

For that we have to calculate P(A),P(B) and P(A Π B)

Sample space for A:{(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

Total outcomes= 36

$P(A) = \frac{6}{36} \\ \\ = \frac{1}{6}$

Sample space for B:{(1,2),(2,2),(3,2),(4,2),(5,2),(6,2)}

Total outcomes= 36

$P(B) = \frac{6}{36} \\ \\ = \frac{1}{6}$

Sample space for A Π B={(6,2)}

$P(A \cap \: B) = \frac{1}{36}$

Now

$P(A).P(B) = P(A \cap \: B)\\\\ \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \\ \\ \frac{1}{36} = \frac{1}{36}$

So, both the events are independent.

Hope it helps you.

Answer 2

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Let us consider two independent events A and B, then P(A ∩ B) = P(A). P(B)

when an unbiased die is thrown twice

S = {(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)

(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)

(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)

(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)

(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)

(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)}

Let us describe two events as

A: odd number on the first throw

B: odd number on the second throw

To find P(A)

A = {(1, 1), (1, 2), (1, 3), …, (1, 6)

(3, 1), (3, 2), (3, 3), …, (3, 6)

(5, 1), (5, 2), (5, 3), …, (5, 6)}

Thus, P (A) = 18/36 = 1/2

To find P(B)

B = {(1, 1), (2, 1), (3, 1), …, (6, 1)

(1, 3), (2, 3), (3, 3), …, (6, 3)

(1, 5), (2, 5), (3, 5), …, (6, 5)}

Thus, P (B) = 18/36 = 1/2

A ∩ B = odd number on the first & second throw = { (1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}

So, P(A ∩ B) = 9/36 = 1/ 4

Now, P(A). P(B) = (1/2) × (1/2) = 1/4

As P(A ∩ B) = P(A). P(B),

Hence, the two events A and B are independent events

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Hope it's Helpful.....:)