# Probability to win in less than 3 coin flips (probability $0.2$ on each flip)

Here's a problem and I have a small doubt.

We flip a coin. We get a head with probability 0.2 on each flip. Each flip is independent. If the player wins (head), he stops. If he loses, he plays again.

What is the probability of the event H: win in less than 3 flips ?

I know that I can answer the following way:

Let G: Getting at least 3 tails.

$P(H)=1 - P(G)=1 - 0,8*0,8*0,8 = 0,488$ which is the right answer.

BUT, we could also say: $P(H) = P(H_1) + P(T_1H_2) = 0,2 + 0,8*0,2 = 0,36$ ?????????

Why is my second result different from the first one please?

Can anyone help ?

Regards,

• The first computes the probability of wining in $\leqslant 3$ flips, the second computes the probability of winning in $< 3$ flips. – Daniel Fischer Sep 4 '13 at 21:40

Look carefully at the wording of the problem. Is it $3$ flips or less or is it less than $3$ flips?

The probability of winning in $3$ flips or less is $1-(0.8)^3$. It is $1$ minus the probability of $3$ tails in a row.

The probability of winning in less than $3$ flips is not $1-(0.8)^3$, it is $1-(0.8)^2$.

Now let's do it your way. You calculated correctly the probability of winning in less than three flips. You got $0.36$, and as you can see, that is the same as $1-(0.8)^2$.

Using your procedure, the probability of winning in $3$ flips or less can be computed as $0.2+(0.8)(0.2)+(0.8)^2(0.2)$. That gives the same result as $1-(0.8)^3$.

• André, winning in less than 3 flips is winning in one or two flips, isn't it ? It does not mean winning in 3 flips or less ? This is the thing I cannot figure out. – XCoder Sep 4 '13 at 21:50
• Winning in less than 3 flips, does not mean in 3 flips or less, but the solution is 0,488. This looks really strange to me. – XCoder Sep 4 '13 at 21:52
• That's right. Winning in less than $3$ flips means $1$ or $2$. But then $1-(0.8)^3$ is wrong, it should be $1-(0.8)^2$, which agrees with your calculation. – André Nicolas Sep 4 '13 at 21:53
• Oh you are reassuring me. But in this case, the result should be 0,36 but it is not in the possible results (multiple choice) of the question. But 0,488. Does that mean that the multiple choice is wrong ? – XCoder Sep 4 '13 at 21:55
• If it says, verbatim, "less than three flips" then $0.36$ is right and $0.488$ is wrong. – André Nicolas Sep 4 '13 at 22:00