Answer to Question #131905 in Statistics and Probability for aman

Question

suppose that an airplane engine will fail, when in flight, with probability 1−p independently from engine to engine; suppose that the airplane will make a successful flight if at least 50 percent of its engines remain operativ

Answer 1

Here given,

engine fail probabilityP(F) = 1−p independently from engine to engine.

engine success probability P(S) = 1-(1-p)

= p

There no question specified . I think a question is like

'For what values of p is a four-engine plane preferable to a two-engine plane?'

For making four engine airplane success at least 2 engine of them should be success.

so, probability = $(4C2)p^2(1-p)^2 + (4C3)p^3(1-p)^1 + (4C1)p^4(1-p)^0 = 6p^2(1-p)^2+4p^3(1-p) + 4p^4$

required Probability for 2 engine airplane success at least 1 engine should success

= $(2C1)p^1(1-p)^1 + (2C2)p^2(1-p)^0$

=2(p)(1-p) + p²

The four-engine plane is safe if:

 $6p^2(1-p)^2+4p^3(1-p) + 4p^4 \geq 2(p)(1-p) + p^2$

$6 p(1− p) + 4 p (1− p) + p ≥ 2 − p$

$3p^3-8p^2+7p-2 \geq0 \,or \, ( p-1)^2 (3p-2) \geq 0$

$3p-2 \geq 0 \,so, \, p \geq2/3$