A coin can generate 1/5 chance?

Discrete Mathematics Level 5

You have a fair coin. Can you use this coin to generate a random event with \(\frac{1}{5}\) chance of success?

Of course you can! For example, if you toss the coin 3 times, you get eight possible tosses HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. You can designate the first (HHH) as a success, the next four (HHT, HTH, HTT, THH) as a failure, and the remaining three indicates that you redo it all over again.
We can prove that the expected number of coin tosses used in this method is \( 3 \times \frac{1}{ 1 - \frac{5}{8} } = \frac{24}{5}\).

Find the strategy that minimizes the expected number of coin tosses that are needed. The expected value can be expressed as \(\frac{a}{b}\), where \(a,b\) are positive integers that are relatively prime. Enter your answer as \(a+b\).


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