Roll a die until a 6, P(first 6 within 3 rolls)

Step by step

The question

Question: we roll a die until we get a 6: what is the probability that the first 6 comes within 3 rolls?

1 Name p and q

p, the chance of a success on one try, is 1/6.

q, the chance of a fail, is 1 minus p: 5/6.

Every try has the same chance, and one try does not change the next.

2 None in 3 rolls

Within 3 means on roll 1, or 2, up to 3.

The only other way: all 3 rolls fail.

All 3 fail: q times itself 3 times.

That is 125/216, about 0.5787.

3 The rest

Everything else has a success within 3: 1 minus that is about 0.4213.

Careful!

p to the power 3 would be 3 successes in a row.

We only need one.

4 Check

Let's check by adding each roll: 0.1667 plus 0.1389 plus 0.1157.

That is about 0.4213.

So it is right!

Let's look back.

First: name p and q.

Then: none in 3 rolls.

Then: the rest.

Last: check.