Roll a die until a six, earn five dollars per roll

Question

You roll a fair six-sided die over and over until the first time it shows a 6, then you stop. You earn 5 for every roll you make, including the roll that shows the 6. On average, how much do you earn?

Accepted Answer

Earnings are 5 per roll, so find the average number of rolls first, then multiply. Average number of rolls. Each roll shows a 6 with probability 1/6. The waiting-time rule says: if an event has probability p on each try, the average number of tries until it first happens is 1/p. Here p = 1/6, so the average number of rolls is: \frac{1}{1/6} = 6. This matches intuition: a 6 comes up one time in six on average, so you should expect to roll about six times before you finally see one. Average earnings. At 5 per roll: EV = 5 	imes 6 = 30. Same idea as the coin games, just with a die: rarer target…

Roll a die until a six, earn five dollars per roll

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