Question

What is the probability of getting exactly `2` three's when rolling `4` fair, `6`-sided die?

I need a solution that isn't `(4c2)*(1/6)^2*(5/6)^2`

I need a casework type answer.

Answer 1

The solution is indeed `""_4C_2 (1/6)^2 (5/6)^2`. The approximate decimal value for this is `0.1157.`

Explanation:

This question uses a binomial distribution—repeated independent trials, each with only two options (success or failure).

For each die, the probability of getting a 3 is `1/6`. (Hence, the probability of getting anything NOT 3 is the remaining `5/6`.)

We are interested in the probability of getting exactly 2 threes. This means the other two dice show anything that's NOT 3.

Let's imagine the dice are rolled one at a time. What's the probability of rolling the two 3's on the first two rolls? Since all 4 rolls are independent, this would be

`"P"(3) * "P"(3) * "P"("not 3") * "P"("not 3")`

`=1/6 * 1/6 * 5/6 * 5/6`

`=(1/6)^2(5/6)^2`

But wait—this can't be our final answer, because there are other orders in which we could roll two 3's. (i.e. 3x3x, 3xx3, etc.)

To make this right, we need to account for all the orders in which we could get two 3's out of four rolls. This is where the `""_4C_2` part comes in. We have 4 rolls, and we're choosing 2 of them to be successes. There are `""_4C_2=6` ways to do this:

33xx, 3x3x, 3xx3, x33x, x3x3, xx33

To get the total probability for two 3's in any order, we add the probabilities of each of these orders together. Since they all involve independently rolling two 3's and two NOT 3's, each order has the same probability of `(1/6)^2(5/6)^2,` and we simply multiply that one probability by how many orders there are (that being `""_4C_2`, or 6).

This is where we get `""_4C_2 (1/6)^2 (5/6)^2` from.

While many probability tests will allow you to stop here (because it shows you know how to arrive at the answer), you can easily multiply this out to get a proper fraction, and then an approximate decimal:

`""_4C_2 (1/6)^2 (5/6)^2 = 6(1/36)(25/36)`
`color(white)(""_4C_2 (1/6)^2 (5/6)^2) = 25/216`
`color(white)(""_4C_2 (1/6)^2 (5/6)^2) ~~ 0.1157" "= 11.57%`