How long does it take you to roll a Yahtzee?

The dice game Yahtzee is a classic. Each player takes turns rolling 5 dice, 3 times each. After each roll, the player can set aside some dice that they want to "keep" and reroll the rest in order to get a better score. The best roll that someone can have is called a "Yahtzee," or 5 dice of the same number, e.g. five 1s, five 2s, etc.

It's easy to determine the probabilities of getting a Yahtzee in 1 roll. You have 5 dice, and a dice has 6 sides, so the chance of getting all 5 the same is \(\left(\frac{1}{6}\right)^5\), but since there are 6 ways to get a Yahtzee, the probability is really \(\left(\frac{1}{6}\right)^4 = \frac{1}{1296}\). From this, we can determine that one should expect to roll 5 dice 1296 times before getting a Yahtzee, right?

If you've ever played, you'll realize that the chances of getting a Yahtzee in a game are higher than that. This is because of the rule that you can keep some dice aside after each roll. If after each roll you decide to keep the dice aside that have the number that shows up the most, given an infinite number of rolls (not limited to 3 rolls as in real games), what is the expected number of turns it should take you to get a Yahtzee?

Round your answer to the nearest whole number.

Example sequence of rolls:

  • Roll 1: 1, 2, 4, 4, 5. Keep the two 4s.

  • Roll 2: 4, 4, 6, 3, 3. Keep the two 4s.

  • Roll 3: 4, 4, 1, 1, 1. Keep the three 1s.

  • Roll 4: 1, 1, 1, 5, 1. Keep the four 1s.

  • Roll 5: 1, 1, 1, 1, 1. Done after 5 rolls.


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