(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

An unbiased six-sided dice is thrown 10 times. What is the probability that exactly 4 of any one number alone will occur?

2. Relevant equations

Binomial equation

Combinatorial equations

3. The attempt at a solution

You would be interested in the probability of choosing any number 4 times out of the 10 throws:

6*(10 C 4) (1/6)^4 * (5/6)^6

You would also be interested in the chance of throwing that same number that has already been thrown 4 times a 5th, 6th, 7th, 8th, 9th and 10th time (since that side of the dice isn't removed, it could still come up in the remaining 6 throws).

...(6 C 1) (1/6)^1 * (5/6)^5

+ (6 C 2) (1/6)^2 * (5/6)^4

+ (6 C 3) (1/6)^3 * (5/6)^3

+ (6 C 4) (1/6)^4 * (5/6)^2

+ (6 C 5) (1/6)^5 * (5/6)^1

+ (6 C 6) (1/6)^6

And also the chance that one of the remaining 5 numbers could be thrown 4 times...

5 * (6 C 4)*(1/6)^4 * (5/6)^2

However, these last two sets of equations aren't entirely independent of one another...you could, for example, start by throwing four 3's...and then in your remaining 6 tosses you get another 3 and then four 5's and then a 6.

I can't quite figure out how to separate out these values into truly independent events in order to combine them and get to the answer.

Am I on the right path here or is there a simpler way to think about this that I have overlooked? I'm quite stuck here so any help would be greatly appreciated.

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# Homework Help: Probability question (throwing dice)

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