20180223 Riddler Express

There are two possible end states: I have three tokens (win) or I have no tokens (lose).

We can draw a tree to illustrate the possible outcomes:

Now we can begin summing up the probabilities of the win cases, to see if a pattern emerges. A win by Round 2 would have the probability:

A win at or before Round 4 would have the probability:

A win at or before Round 6 would have the probability:

We have identified an infinite series and can solve as follows:

As a sense check, we can run a simple 1000-trial simulation in Excel to see if it approximates 4/7, or ~57.1%. Using the normal approximation interval at 95% confidence, we would expect to see a result within 3.1%.

Here are our results (dark green is a win, red is a loss):

In this simulation, we saw 577 wins out of 1000 trials, giving us an experimental 57.7% win rate, well within our confidence interval.

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