# Flip a coin 3 times. What is the probability that number of tails is odd?

We flip a fair coin (independently) three times. Define the following events:

A = "the number of tails is odd"

B = "the number of heads is even"

What is the probability of event A and event B?

HINT: $A=\{HHT,HTH,THH,TTT\}$ and $B=\{HHT,HTH,THH,TTT\}$

• Wouldn't 0 heads be valid for B? – RandomUser Apr 25 '14 at 21:34
• So Pr(A)= 4/2^3 ? since total possible outcomes are 2^3? – Andrew Apr 25 '14 at 21:34
• @RandomUser I was just about to suggest this. – homegrown Apr 25 '14 at 21:35
• @RandomUser: Yes you are right. Mistake! – Argha Apr 25 '14 at 21:35
• @jnh Yup, it says I cant accept for 1 more minute. Will do so. – Andrew Apr 25 '14 at 21:44

The probability of an odd number of heads is the same as the probability of an odd number of tails - Prob(H)=Prob(T). And either the number of heads or tails are odd - 2 Prob(T) = 100%.

So 50%.

The same for an even number.

Whichever event you're betting on, A or B, it all comes down to the last flip: if it comes up one way you win, the other way you lose. Assuming it's a fair coin, your probability of winning is 50%.