Mars landing A space vehicle is designed to land on Mars. Assume that the ground

Question

Mars landing A space vehicle is designed to land on Mars. Assume that the ground conditions on Mars are either hard or soft. If hard ground (H) is encountered during landing, the probability of successful landing is P (L | H) = 0.9, while the probability of successful landing on a soft ground (H bar) is only P(L | H bar) = 0.5. Based on available information, the chance of landing on hard ground is three times the probability of landing on soft ground. a) What is the probability of successful landing? b) Suppose a stick can be projected to test the ground conditions before landing. It will penetrate into soft ground with P (P | H bar) = 0.9 probability and hard ground with P(P | H) = 0.2 probability. If the stick was observed to penetrate into the ground, what is the poissibility that the ground is hard?

Explanation / Answer

H: Hard ground

L: Landing

P(L|H)=0.9 , P(L|notH)=0.5 also, it is given that P(H)=3*P(notH) and we know that P(H)+P(notH)=1 thus P(H)=1-P(H)/3 , or P(H)=3/4 and P(notH)=1/4

a) P(L)=P(H)*P(L|H)+P(notH)*P(L|notH) = (3/4)*0.9+(1/4)*0.5 =0.8

b) P: stick penetration

P(P|notH)=0.9 and P(P|H)=0.2

Thus P(P)=P(H)*P(P|H)+P(notH)*P(P|notH) =(3/4)*(0.2)+(1/4)*(0.9)= 0.375

Using Baye's theorem, P(H|P)=P(P|H)*P(H)/P(P)=0.2*(3/4)/0.375 =0.4

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