Flip a coin until tails, P(first tails within 3 flips)

Step by step

The question

Question: we flip a coin until we get tails: what is the probability that the first tails comes within 3 flips?

1 Name p and q

p, the chance of a success on one try, is 1/2.

q, the chance of a fail, is 1 minus p: 1/2.

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

2 None in 3 flips

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

The only other way: all 3 flips fail.

All 3 fail: q times itself 3 times.

That is 1/8, about 0.1250.

3 The rest

Everything else has a success within 3: 1 minus that is about 0.8750.

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 flip: 0.5000 plus 0.2500 plus 0.1250.

That is about 0.8750.

So it is right!

Let's look back.

First: name p and q.

Then: none in 3 flips.

Then: the rest.

Last: check.