Waiting for a satellite's first successful ping

Question

A weather satellite tries to contact a ground station exactly once per orbit. Each attempt succeeds independently with probability $0.4$ , and it keeps trying on later orbits until one gets through.

On average, how many orbits pass until the first successful contact (counting the orbit that succeeds)?

Accepted Answer

This is the classic "wait for the first success" setup. Each orbit is an independent trial that succeeds with probability p = 0.4. Let E be the expected number of orbits until the first success. Think about the very first orbit. It costs you 1 attempt no matter what. With probability p it already succeeds and you are done. With probability 1-p it fails, and you are right back where you started, still needing an expected E more orbits. So E = 1 + (1-p)\,E. Solve for E: E - (1-p)E = 1 \quad\Longrightarrow\quad pE = 1 \quad\Longrightarrow\quad E = \frac{1}{p}. With p = 0.4, E = \frac{1}{0.4} = 2…

Waiting for a satellite's first successful ping

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