# What is the difference between these two probability problems?

davedave
I happened to come across these two probability questions in a library book.

1) Suppose that you roll 5 six-sided dice at the same time. What is the probability of getting
a sum of 20?

2) Suppose that you roll a six-sided die 5 times. What is the probability of getting a sum of
20 on the 5 rolls?

What is the difference between these two problems?

What methods are used to solve them?

bpet
What is the difference between these two problems?

Good question. If the dice are identical then the final answer will be the same because the order of the dice rolls doesn't affect the outcome of the sum. The event space of dice rolls can be written as {1,2,...,6}^5 = {(1,1,1,1,1),(1,1,1,1,2),...,(6,6,6,6,6)} with all events equally probable, so you'll need to find a way of counting the number of events with sum 20.

awkward
Good question. If the dice are identical then the final answer will be the same because the order of the dice rolls doesn't affect the outcome of the sum. The event space of dice rolls can be written as {1,2,...,6}^5 = {(1,1,1,1,1),(1,1,1,1,2),...,(6,6,6,6,6)} with all events equally probable, so you'll need to find a way of counting the number of events with sum 20.
An easy way (given modern technology) of counting the number of events that sum to 20 is to observe that it is the coefficient of x^20 in (x + x^2 + x^3 + x^4 + x^5 + x^6)^5.

Just type "expand (x + x^2 + x^3 + x^4 + x^5 + x^6)^5" into Wolfram Alpha and hit Enter.

You can also find the coefficient of x^20 by paper and pencil methods-- it's not that hard, just a little more work.

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