Question

A coin that lands on heads with probability p is placed on the ground, showing heads, at timet 0. Thereafter, randomly but with a rate of λ times per hour, the coin is picked up and flipped. (a) What is the probability that the coin shows heads at any time t? (b) Suppose that instead of flipping it, we pick the coin up and turn it over. What is the probability that the coin shows heads at any time t? Hint: If Nnumber of flips until a given time t, then what is the distribution of N? Then for (a), condition on the event N0

Answer 1

Number of flips N_t until a given time t (hours) is a poisson process with mean $\lambda$ per hour.
$P(N_{t}=k) = e^{-\lambda t}\frac{(\lambda t)^{k}}{k!}$
Probability of heads, P(H) = p
Probability of heads at first flip = p
Probability of heads at 2nd flip when coin is flipped 2 times = (1-p)p + p² = p
Probability of heads at nth flip when coin is tossed n times = p

(a)
Probability that coin shows heads at any time t:
P(N_t=0) + P(N_t=1).P(H) + P(N_t=2)P(Heads at 2nd flip) + P(N_t=3)P(Heads at 3rd flip) + ... P(N_t=k)P(Heads at nth flip)
$=e^{-\lambda t}\frac{(\lambda t)^{0}}{0!} + e^{-\lambda t}\frac{(\lambda t)^{1}}{1!}p +e^{-\lambda t}\frac{(\lambda t)^{2}}{2!}p +...$
$=e^{-\lambda t}(\frac{(\lambda t)^{0}}{0!} + \frac{(\lambda t)^{1}}{1!}p +\frac{(\lambda t)^{2}}{2!}p +...)$
$=e^{-\lambda t}(1 + \frac{(\lambda t)^{1}}{1!}p +\frac{(\lambda t)^{2}}{2!}p +...)$
(b)
Probability that coin shows heads at any time t:
On every even shift heads will show up.
P(N_t=0) + P(N_t=2) + P(N_t=4) + ...
$=e^{-\lambda t}\frac{(\lambda t)^{0}}{0!} + e^{-\lambda t}\frac{(\lambda t)^{2}}{2!} +e^{-\lambda t}\frac{(\lambda t)^{4}}{4!} +...$
$=e^{-\lambda t}(1 + \frac{(\lambda t)^{2}}{2!} +\frac{(\lambda t)^{4}}{4!} +...)$