## The answer for the previous challenge

This is the question in my last post:
If we have two coins which one is double headed, and we toss a coin and is head, what is the probability for the other side to be a tail?

If your answer is 1/2, then you are wrong (as I was). I thought this is the same problem as: what is the probability of a selected coin being a normal coin which is 1/2.

I was told that the answer is 1/3 and for some time I was convinced that 1/3 is the right answer but after thinking more about the problem. I thought again in calm and again realized how naive I was the first time.

To understand the problem better let’s rephrase it as below:
We have two boxes one contains a pair of black beans and the other contains a black bean and a white bean. We pick a bean from a box and it is black. What is the probability of “picking another bean from the same box and it being white”?
Look at the figure below: After taking a black bean out, the bean might be either from the box with  white bean in it or from the box with black bean in it or again from the box with another black bean in it.

Sometimes formulas come to rescue.

P(H) is the probability of getting the Head. And P(H) = 1 – P(T). There is no conditional probability here. If we translate the problem as: Having a head out, what is the probability of NOT having another head.

Having a head out P(H) = 2/3 and P(T)=1-P(H)=1/3.

### One Response to “The answer for the previous challenge”

1. Andy Says:

The reason that the probability is 1/3 is that there are actually 3 possible outcomes not 2 as you have drawn in your diagram.

You have drawn the two outcomes
1. Select the black bean in the container with two black beans
2. Select the black bean in the container with a black and a white bean.

So the chance of pulling a white bean is ½ but…

there are really three possible outcomes
1. Select the first black bean in the container with two black beans
2. Select the second black bean in the container with two black beans.
3. Select the black bean in the container with a black and a white bean.

So the chance of pulling a white bean is 1/3

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You are absolutely right Andy. I fixed the post.