How do i work with this problem for stats?

Suppose an airplane engine will fail (when the plane is in flight) with probability . Failure of engines are statistically independent events. Suppose, also, that the plane will land successfully if at least half of its engines remain operative. For what values of p is a 4-engine plane preferred to a 2-engine plane (i.e., for what values of p is the probability of a successful landing higher with a 4-engine plane)?How do i work with this problem for stats?
p is the probability that an engine will fail.


For a two engine plane the maximum number engine failures so that the plane lands safely is 1. That is, the number of failures can be either 0 or 1.





Let x represent the random variable 'number of failures'.





P(x 鈮?1) = P(x = 0) + P(x = 1) = C(2,0)p掳(1鈹€p)虏 + C(2,1)p(1鈹€p)


= 1 + p虏 鈹€ 2p + p 鈹€ p虏 = 1 鈹€ p





For a four engine plane the maximum number engine failures so that the plane lands safely is 2. That is, the number of failures can be either 0 or 1 or 2.





P(x 鈮?2) = P(x = 0) + P(x = 1) + P(x = 2)


= C(4,0)p掳(1鈹€p)^4 + C(4,1) p(1鈹€p)^3 + C(4,2)p虏(1鈹€p)虏





= 1 鈹€ 4p + 6p虏 鈹€ 4p^3 + p^4 + 4(p 鈹€ 3p虏 + 3p^3 鈹€ p^4 )


路 路 路 + 6(p虏 鈹€ 2p^3 + p^4)





= 1 鈹€ 4p^3 + 3p^4





The above probabilities are the probabilities of less than half the number engines of the plane, to fail. So, four engine planes are preferred to two engine planes when





1 鈹€ 4p^3 + 3p^4 %26gt; 1 鈹€ p





That is,


3p^4 鈹€ 4p^3 + p %26gt; 0





p(3p^3 鈹€ 4p虏 + 1) %26gt;0





p(p 鈹€ 1)(3p虏 鈹€ p 鈹€ 1) %26gt; 0





Also, p 鈺?0, 1


0 鈮?p 鈮?1 always. Hence, p(p 鈹€ 1) 鈮?0, which implies


3p虏 鈹€ p 鈹€ 1 %26lt; 0


This implies 鈹€ 0.4343 %26lt; p %26lt; 0.7676





But p %26gt; 0


Hence for 0 %26lt; p %26lt; 0.7676, four engine planes are preferred to two engine planes