A die is rolled 20 times and the number of twos that come up is tallied.?

Question

A die is rolled 20 times and the number of twos that come up is tallied. Find the probability of getting the given result.

Exactly five twos.

Thanks!

Answer 1

This problem kind of sucks. Here we go:

First, note that in probability problems, in general, x means "and," while + means "or."

Now, on any given roll, the probability of rolling a 2 is 1/6.
(duh--we're using a fair die with 6 numbers).

So the probability of rolling anything else is 1-1/6=5/6.

So, we are rolling a die 20 times.

We need to get a 2 exactly 5 times, which means we need to get something other than a 2 exactly 15 times.

Then, of exactly 20 rolls, we need 5 to be 2. However, note that these can be any 5 rolls, not necessarily the first 5 or the last 5 or any other specific order. So order doesn't matter and we use a combination, and since we're taking 5 rolls out of 20, that would be 20C5.

So putting to gether these three parts yields:

20C5 x (1/6)^5 x (5/6)^15

Multiplying that out yields the right answer.

Answer 2

P(Five 2s) = (20C5) [(1/6)^5] [(5/6)^15]

= 15504*0.000128601*0.064905472 = 0.129410292