Question

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is ______.

Options

• `3/16`

• `1/2`

• `5/16`

• `1/32`

Answer 1

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is `underlinebb(1/2)`.

Explanation:

Given: An ordinary dice is rolled a certain number of times. The probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times.

Probability of r successes in n trials = ⁿC_rp^rq^n–r

Where p = Probability of success

q = 1 – p = Probability of failure

p = Probability of getting odd number

= `3/6` = `1/2`

q = Probability of getting even number

= `1 - 1/2` = `1/2`

P(getting odd number twice) = P(getting even number thrice)

⇒ `""^nC_2(1/2)^2(1/2)^(n-2) = ""^nC_3(1/2)^3(1/2)^(n-3)`

⇒ ⁿC₂ = ⁿC₃

⇒ n = 2 + 3

⇒ n = 5

P(getting odd number for odd number of times) = P(odd no. 1 times)+ P(odd no. 3 times) + P(odd no. 5 times)

= `""^5C_1(1/2)^5 + ""^5C_3(1/2)^5 + ""^5C_5(1/2)^5`

= `16/2^5` = `1/2`.