Probability of 5 dice rolled

[Susan0814]

When rolling 5 dice one time for each, what is the formula to figure the probability of getting a 5 of a kind, 4 of a kind, 3 of a kind, pair, and a straight.
I used 6*(1/6*1/6*1/6*1/6*1/6) to get the probability of 5 of a kind and came up with 6/7776. What formulas do I use to get the others or how do I alter the formula?

 

[Dick]

You are off to a good start. The formula is (how many kinds)*(probability of getting n of the kind)*(how many ways to get the n). You've correctly done the five of a kind as 6*(1/6)^5*1. Now let's do 4. The first number is the same. The second number becomes (1/6)^4*(5/6) (four must be the selected number and the fifth can't). The last number is how many ways can this happen. Any of the 5 numbers can be nonmatching so 5. So 6*(1/6)^5*(5/6)*5. You can creatively make these up for each case, or you can think of some general formula you've been taught to apply. Hint: one of the numbers is called a combinatorial coefficient.

 

[Architeuthis Dux]

To calculate the probability of getting a 4 of a kind, 3 of a kind, or a pair, you can use the following formula:

Number of ways to get the desired outcome / Total number of possible outcomes

For a 4 of a kind, there are 6 ways to choose the number that will appear 4 times, and 5 ways to choose the remaining die. This gives us a total of 6*5 = 30 ways to get a 4 of a kind. The total number of possible outcomes when rolling 5 dice is 6^5 = 7776. So the probability of getting a 4 of a kind is 30/7776 or approximately 0.00386.

For a 3 of a kind, there are 6 ways to choose the number that will appear 3 times, and 5C2 = 10 ways to choose the remaining two numbers. This gives us a total of 6*10 = 60 ways to get a 3 of a kind. So the probability of getting a 3 of a kind is 60/7776 or approximately 0.00772.

For a pair, there are 6 ways to choose the number that will appear twice, and 5C3 = 10 ways to choose the remaining three numbers. This gives us a total of 6*10 = 60 ways to get a pair. So the probability of getting a pair is 60/7776 or approximately 0.00772.

To calculate the probability of getting a straight, you can use the following formula:

Number of ways to get a straight / Total number of possible outcomes

There are 6 possible straight combinations (12345, 23456, 34567, 45678, 56789, 67891) and each can appear in 5 different ways (since the first die can be any number). So the total number of ways to get a straight is 6*5 = 30. The total number of possible outcomes when rolling 5 dice is 6^5 = 7776. So the probability of getting a straight is 30/7776 or approximately 0.00386.

In summary, the formulas to calculate the probability of getting a 5 of a kind, 4 of a kind, 3 of a kind, pair, and a straight when rolling 5 dice are as follows:

5 of a kind