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प्रश्न
The probability that a mountain-bike travelling along a certain track will have a tyre burst is 0.05. Find the probability that among 17 riders: two or more have burst tyre.
उत्तर
Let X = number of burst tyre.
p = probability that a mountain-bike travelling along a certain track will have a tyre burst
∴ p = 0.05
∴ q = 1 - p = 1 - 0.05 = 0.95
Given: n = 17
∴ X ~ B(17, 0.05)
The p.m.f. of X is given by
P(X = x) = `"^nC_x p^x q^(n - x)`
i.e. p(x) = `"^17C_x (0.05)^x (0.95)^(17 - x)`, x = 0, 1, 2,...,17
P(two or more have tyre burst)
= P(X ≥ 2) = 1 - P(X < 2)
= 1 - [P(X = 0) + P(X = 1)]
`= 1 - [p(0) + p(1)]`
`= 1 - [""^17C_0 (0.05)^0 (0.95)^17 + "^17C_1 (0.05)^1 (0.95)^16]`
`= 1 - [1(1)(0.95)^17 + 17(0.05)(0.95)^16]`
`= 1 - (0.95^16) [0.95 + 0.85]`
`= 1 - (1.80)(0.95)^16`
`= 1 - (1.8)(0.95)^16`
Hence, the probability that two or more riders have tyre burst = `1 - (1.8)(0.95)^16`.
Notes
The answer in the textbook is incorrect.
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