# 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. - Mathematics and Statistics

Sum

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.

#### Solution

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

[Note: Answer in the textbook is incorrect.]

Concept: Binomial Distribution
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 8 Binomial Distribution
Miscellaneous exercise 8 | Q 5.3 | Page 254