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# A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at most 5 successes. - Mathematics and Statistics

Sum

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at most 5 successes.

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#### Solution

Let X = number of successes, i.e. number of odd numbers.

p = probability of getting an odd number in a single throw of a die

∴ p = 3/6 = 1/2 and q = 1 - "p" = 1 - 1/2 = 1/2

Given: n = 6

∴ X ∼ B(6, 1/2)

The p.m.f. of X is given by

p("X = x") = "^nC_x  p^x  q^(n - x)

i.e. p(x) = "^6C_x (1/2)^x (1/2)^(6 - x)

 = "^6C_x (1/2)^6, x = 0, 1, 2, ...,6

P(at most 5 successes) = P[X ≤ 5]

= 1 - P [X > 5]

= 1 - p(6) = 1 - "^6C_6 (1/2)^6

= 1 - 1 xx 1/64 = 63/64

Hence, the probability of at most 5 successes is 63/64.

Concept: Binomial Distribution
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