Probability and Yahtzee

[silvermane]

Homework Statement

What is the probability of rolling a yahtzee, where 1 turn consists of 3 rolls, where the player may pull aside the dice that help with getting the 5 of a kind, and the other dice may be rolled again until a total of 3 rolls have been made.

Homework Equations

probability = #of outcome in subset a divided by total outcomes in S.

The Attempt at a Solution

I do know that if I were to re-roll all of the dice until the outcome we need, we can get 6 divided by 6^5. I don't know how to calculate it where it's depending on what you get in the first roll. I think a matrix is needed though; any suggestions?

 

[mighty2000]

The probability of rolling a yahtzee in one turn can be calculated as follows:

First, we need to calculate the total number of possible outcomes in one turn, which is 6^5 (since there are 5 dice and each die can have 6 possible outcomes).

Next, we need to calculate the number of outcomes that result in a yahtzee. This can be done by considering the different scenarios:

1. The first roll results in a yahtzee: This has a probability of (1/6)^5, since all 5 dice need to land on the same number.

2. The first roll does not result in a yahtzee, but one of the dice is the same as the yahtzee number: This has a probability of (5/6) * (1/6)^4, since one of the dice needs to land on the yahtzee number and the remaining 4 dice can have any outcome.

3. The first roll does not result in a yahtzee, and none of the dice are the same as the yahtzee number: This has a probability of (5/6)^5, since all 5 dice need to land on a different number.

Thus, the total number of outcomes that result in a yahtzee is (1/6)^5 + (5/6) * (1/6)^4 + (5/6)^5.

Therefore, the probability of rolling a yahtzee in one turn is (1/6)^5 + (5/6) * (1/6)^4 + (5/6)^5 divided by 6^5.

I hope this helps! Let me know if you have any further questions.