# Probability and Expectation

1) A coin having probability p of coming up heads is continually flipped until both heads and tails have appeared. find (a) the expected number of flips (b) the probability that the last flip lands heads

2) Ten hunters are waiting for ducks to fly by. when a flock of ducks flies overhead, the hunters fire at the same time, but each chooses his target at random, independently of the others. if each hunter independently hits his target with probability 0.6, compute the expected number of ducks that are hit. Assume that the number of ducks in a flock is a Poisson random variable with mean 6.

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#### Solution Preview

1) Probability of getting head = p, probability of getting tail = q.

The the random variable follows Geometric distribution with pdf given as:

P(n) = p(1-p)n = qnp

[if p = ½ = q]

then P(n) = (½)n(1/2) = (1/2)n+1

=> P(n) = (½)n+1

Expected value = nP(n) = nqnp

n=0

E(X) = p[0q0 + 1q1+2q2 + ..... ] = p [1q + 2q2 + 3q3 + ......]

E(X) ...

#### Solution Summary

The solution explains the probability expectations. The solution is step by step with tables and equations.