MATHS M1 - Ans.fyi 參考答案

MATHS M1

2012-06-30 8:16 pm

A rocket is projected by a system of 2 large engines or a system of 4 small engines.The rocket can get into orbit as long as half of the engines work.Tests on the engines indicate that the probability of failure of a mall engine is double that of a large engine.Let p be the probability of failure of a large engine.

(a) let X be the number of engines that fail in the large engine system.Express the probability that a system of large engines fails in terms of p.

(b) let Y be the number of engines that fail in the small engine system.Express the probability that a system of small engines fails in terms of p.

(c) if both systems have an equal chance of failure , find the probability that a large engine fails.

回答 (1)

胡雪8°

2012-07-01 12:22 am

✔ 最佳答案

(a)
Consider the large engine system (2 large engines) :
P(a large engine fails) = p
P(a large engine works) = 1 - p

X : the number of large engines that fail

P(a system of large engines fails)
= P(X > (2/2))
= P(X > 1)
= P(X = 2)
= p^2

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(b)
Consider the small engine system (4 large engines) :
P(a small engine fails) = 2p
P(a small engine works) = 1 - 2p

Y : the number of small engines that fail

P(a system of large engines fails)
= P(Y > (4/2))
= P(Y > 2)
= P(Y = 3) + P(Y = 4)
= C(4,3) x (2p)^3 x (1 - 2p) + (2p)^4
= 32p^3 - 64p^4 + 16p^4
= 32p^3 - 48p^4

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(c)
P(a system of large engines fails) = P(a system of large engines fails)
p^2 = 32p^3 - 48p^4
48p^4 - 32p^3 + p^2 = 0
p^2 (48p^2 - 32p + 1) = 0
p^2 = 0 (rejected) or 48p^2 - 32p + 1 = 0
p = [32 ± √(32^2 - 4 x 48 x1)] / (2 x 48)
p = (4 + √13)/12 (rejected)or p = (4 - √13)/12

Probability that a large engine fails
= (4 - √13)/12
≈ 0.03287 (to 4 sig. fig.)

參考: 胡雪

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