On The Roll Again


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All other locations have zero probability. The first few entries are easy to fill in. The probability of moving from state 5 to state 5 is 1. The stochastic nature of the transition matrix is maintained because the row of this matrix sums up to 1.

For state 3 , there are two dice to re-roll, giving 36 possible combinations. Recall back to the section of combinatronics above. Going from state 2 to state 5 requires all three re-rolled dice matching the current pair. Similarly going from nothing matching, state 1 , to state 5 , is the equivalent of rolling a Yahtzee in one go because all the dice are re-rolled. Remember back to the Combinatronics section? If it's not clearly obvious, the probabilities on the top row changing from nothing matching to any other state , are the probabilities for the outcome of the first roll.

This is a little more challenging to calculate, and we have to be careful. Referencing the combinatronics section above remember, I said it would be useful?

Your Answer

To fill in the last element in this row getting a pair , again we have to be careful. Single Pair and Two Pair.

Now the last two tricky ones. Reading a little on the internet, this is where people appear to slip up in their calculations. The complication arises because, when re-rolling three dice, you might want to "jump ship" about what you are aiming for.

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For example, if you rolled a pair on your first roll, kept the pair and re-rolled three dice, and these three dice all came up the same but not the same as the pair, otherwise that would be Yahtzee! This subtlety modifies the probabilities. Similarly, the inverse of this trick for the transition probability from a pair state 2 to three of a kind state 3.

This helps confirm our calculations are correct. We now input an identity column vector and multiply this by the Transition Matrix.

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The resultant row vector output shows the probabilities of the distrubutution of states that the dice could be in. We can now take this output and use it at as the input again and this time the output is the superposition of the probabilities for the dice at the end of the second roll.


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To get the probability of obtaining a Yahtzee on or before three rolls, we multiple one last time and read the result of the firth element state 5 in the final output vector. If you missed your Yahtzee roll, what happens if you rolled again? And again? Below is a chart showing the percentage chance of getting a Yahtzee in n -rolls. The x-axis is the number of rolls, and the y-axis shows the percentage chance.

Often you miss a Yahtzee attempt. What is the breakdown of the probabilities? How likley are you to end up with Four of a kind when shooting for your Yahtzee? This information is easy to obtain by reading the value from the output vector of the Markov Chain for state 4.

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The chance that you'll end up with Four of a kind in three rolls is The table on the left shows the percentage breakdown for the the first 25 rolls. Below is the same data presented in graphical format. Notice that after 9 rolls, the Yahtzee becomes the most likely event, and the chance of ending up with a pair or Singleton rapdily falls to noise. On the first roll, it's most likely you'll get a pair , but after that the chances of end up with just a pair rapidly decreases. You'd have to be incredibly unlucky. You can find a complete list of all the articles here.

Click here to receive email alerts on new articles. Yahtzee Probability Santa brought my kids the game of Yahtzee for Christmas.

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Combinatronics Before delving in the Markov Chain, however, it will benefit us to examine the various ways that combinations of dice can be rolled. Here we go … 2 dice This is the trivial case. We can fill in two more coefficients. Our Transition Matrix is now complete! Show more. Show less. Using the thesaurus. Close What are red words? Close Thesaurus. Synonyms and related words. To reduce something: cut back , cut down , reduce Explore Thesaurus. To remove something: remove , sweep away , take off Setting, controlling and changing prices and costs: bargain , beat down , bottom out

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