Five of a Kind is Hard to Get

The Objective : The reason I chose this project is because I always wanted to know why it was so hard to roll a Yahtzee (five of a kind). My objective was to calculate and then test the probability of getting five of a kind with five dice on a single roll.


My materials included 5 regular six-sided dice, pencil, paper, and a personal computer with Microsoft Excel.

I used mathematical probability to determine the chance of rolling five of a kind.

I then rolled five dice 500 times physically to test the probability.

I then used a Microsoft Excel spreadsheet to simulate 24,000 more random rolls.

I repeated the 24,000 random rolls 42 more times for a total of 1,008,000 rolls.

Finally, I compared the actual results to the predicted probability.


The mathematical probability was 1 in 1,296, or 0.0772%. My actual results on over a million simulated rolls were 0.0773%, almost exactly the expected results.


Five of a Kind turned out to be very hard to get on a single roll. I only got 779 of them in over a million rolls. Also, you shouldn't rely on data from small numbers of tests to test probabilities. I rolled one Yahtzee in 500 rolls or a 0.2% chance. My samples of 24,000 rolls varied from a low of 10 to a high of 27 Yahtzee's, or a 2.7x difference. However, in 1,008,000 rolls the probability was 0.0773% or almost exactly the predicted probability.

This project demonstrates how hard it is to get Five of a Kind on a single roll of five dice.

Science Fair Project done By Paul H. Lego



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