## Want to work with me? Count to 3.

This stumps job applicants all the time (although probably not after this post).

I flip a coin two times. Given that at least one of the flips is heads, what's the probability that both flips are heads.

50% you say?

Sorry, but no. Don't feel bad though (these guys - and this whack job - showed no remorse and they screwed up much worse). Most people think it's 50%. There's no trick involved. It's just a matter of counting. Not to imply that everyone with a brain can count.

Let's think about it. What are the possible combinations of outcomes in two coin flips?

The subtle part comes in the statement "at least one of the flips is heads". So, of the 4 possible choices, how many satisfy our condition?

Now, "what's the probability that both flips are heads"? Another way to ask this is how many times do we get 2 heads?

So given the information ("at least one heads"), we get the outcome once out of a possible three choices. That's 1/3, or 33%.

This one trips up people all the time, even though it's quite simple. I suspect because we rely so heavily on our System 1 thinking, especially in a nerve-wracking interview, that we don't give our System 2 time to formulate a response. We end up blurting out the first thing that comes to us. Someone might think that's a sign of foolishness.

I should mentioned that technically this is a conditional probability question and to answer this you would use a Binomial distribution, with p = 0.5 and n = 2. But of course that's unnecessary.

To really answer the question, you need to know how to count. And that's a minimum requirement for a job.