Discrete Mathematics
# Expected Value

Each of the interactions we see, from carefully balanced see-saws to swinging mobiles of glass windows, has a certain probability of succeeding each time it is run. If each interaction takes the same amount of time, and we wanted to minimize the total amount of time taken to produce this video, how should the stunts be arranged?

An infinite line of stepping stones stretches out into an infinitely large lake.

A frog starts on the second stone.

Every second he takes a jump to a neighboring stone. He has a 50% chance of jumping one stone closer to the shore and a 50% chance of jumping one stone further away from the shore.

What is the expected value for the number of jumps he will take before reaching the first stone (the one closest to the shore)?

\[ \frac{ 5 \sqrt{ 5 + 2 \sqrt{5} } } { 24 } , \]

what is the value of \(N\)?

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