## Conditional Probability

Imagine yourself a tourist lost in a national park in a country in which the natives always give false answers when asked for directions. The natives comprise 3/8 of the visitors of the national park and the rest are tourists(5/8). If asked for directions, tourists give the correct answer in 4/5 cases. Answers to repeated questions are independent, even if you ask the same questuion to the same person.

1) You ask someone whether the exit is east or west. They reply east. What is the probability that the answer is correct?

I used conditional probability
P(C)=P(C and T)+P(C and N)
=P(C|T)P(T)+P(C|N)P(N)
=(4/5 x 5/8) + (0 x 3/8)
= 1/2
according to my solution sheet, this answer is right

2) You ask the same person again and recieve the same reply. What is the probability that it is correct?
I used conditional probability again and binomial distribution

P(C|A1=A2)=P((C and(A1=A2))/P(A1=A2)

Since native always lies P(A1=A2|N)=1 (only one way of getting both answers the same)

P((A1=A2)|T)=17/25 (using binomial, n=2, m=2(both correct) plus m=0(both wrong)
so P(A1=A2)=P((A1=A2)|T)P(T)+P((A1=A2)|N)P(N)
=(17/25 x 5/8) +(1 x 3/8)
=4/5
according to my solution sheet, this answer is right

3)You then ask the same person again(for a third time) and recieve the same reply. What is the probability the answer is correct?

16/35 same method as before using binomial, n=3, m=3 plus m=0
according to my solution sheet, this answer is right

4)you ask again for a fourth time, the answer is still east. Probabilty it is correct?
I got 32/125 using the binomial for n=4, m=4 plus m=0. Think it is correct but confirmation would be appreciated.

5) show that if the fourth answer had been west instead( the first three still being east) the probability that east is correct is 16/17. Explain qualitatively the difference between this result and the result in part 4)

HELP! I think I have to use the geometric distribution but all help is welcome and appreciated.

Thank you