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E-Value

Bill flips a coin that can land heads or tails, and you assign equal probabilities to each, with all flips mutually irrelevant. Bill will flip the coin until he sees a consecutive sequence of tails, tails, tails. What is the e-value of your distribution of the number of flips until this happens?

For example, if Bill flips HHTTHTHTTT, then the number of flips until TTT, equals ten.

Solution Preview

Call t the expected value of the number of trials required to get 3 tails in a row.

Now, expected values are computed by summing products of the probability of an event and the value of the event,
that sum taken over all events.

If we get a head on the first toss (probability 1/2), we are no closer to getting three in a row than we were before.
The number of tosses we need to get 3 tails, for this case, is one larger than t.

So, one term in our summation for the expected value, t, is (recursively)

(1/2) * (t+1)

Of course, we might also have ...

Solution Summary

Bill flips a coin that can land heads or tails, and you assign equal probabilities to each, with all flips mutually irrelevant. Bill will flip the coin until he sees a consecutive sequence of tails, tails, tails. What is the e-value of your distribution of the number of flips until this happens?

For example, if Bill flips HHTTHTHTTT, then the number of flips until TTT, equals ten.

$2.19