Probability Theory Q&A: Rolling 5 Dice to Find Probability of 2 Different Values

[rad0786]

Hello, I have two questions regarding probaility, i was hoping some body can help me out.
I know that this is a "Physics Forum" However, i thought somebody might still beable to help :)


When rolling 5 distinuishable dice, find the probability that there are two different values?

I did this by (6 x 5 x 4)/2 x ( 5 ) = 1800
(2 2 1)

and the probability is 1800/(6^5) = 0.23148

Now, I am wondering WHY do you divide the (6 x 5 x 4) by 2?

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[Nate810]

The reason why you divide the (6 x 5 x 4) by 2 is because you are counting each possible combination twice. For example, if you rolled a 1, 2, 3, 2, and 1, then you have two different values (1 and 2). However, this would be counted twice in your calculation since the ordering of the dice does not matter. Therefore, you must divide by two to avoid double-counting the same combination.

 

[quicknote]

Hello,

Thank you for your question. In probability theory, the formula for finding the probability of an event is P(E) = (# of favorable outcomes) / (# of total outcomes). In this case, the favorable outcomes are when there are two different values on the 5 dice, and the total outcomes are all the possible combinations of the 5 dice, which is 6^5 = 7776.

In order to find the number of favorable outcomes, we need to consider all the possible combinations of the two different values on the 5 dice. This can be done by first choosing the two values (6 x 5 = 30 possible combinations) and then arranging them on the 5 dice (5 ways to arrange 2 values = 5). However, this method will result in double counting some combinations, as the order of the two values does not matter. For example, rolling a 1 and then a 2 is the same as rolling a 2 and then a 1.

To account for this double counting, we divide the total number of arrangements by the number of ways the two values can be ordered, which is 2 (because there are two values). This is why we divide by 2 in the formula (6 x 5 x 4)/2.

I hope this explanation helps you understand why we divide by 2 in this situation. If you have any further questions, please let me know.