Green or Red M&M?

A recent post on our Instagram asked a probability question about M&Ms. Here is the reel:

inquire.ed

A post shared by Inquire | Learn to Think (@inquire.ed)

Here is the question in text form:

Two of your friends, Gloobo and Moobo have been given a jar of M&Ms each.

Both the jars contain only two colors of M&Ms, red and green.

Exactly half the M&Ms in each jar are red and exactly half are green.

Both reach into their jars and randomly pull out an M&M.

You are told that one of them pulled out a green M&M. What is the probability that the other also pulled out a green M&M?

Most people, when presented with this question, answer 50%. But, that is not the case. Let us take an alternative question:

An alternative question

Suppose the following:

You are told that one of them pulled out a green M&M. What is the probability that the other also pulled out a green M&M?

was replaced by: You are told that Gloobo pulled out a green M&M. What is the probability that Moobo pulled out a green M&M?

In this case, the probability is 50% since there is no relationship between Gloobo pulling out a green M&M and Moobo pulling out a green M&M from a different jar.

Listing possibilities

The difference between the two questions is that in the original, we don’t know who has pulled out a green M&M, while in the second we know that Gloobo pulled one out.

Let us list the possibilities for Gloobo and Moobo pulling out M&Ms.

Rows 2-4 above represent the cases where one of them pulled out a green M&M. So, let us restrict ourselves to those cases:

Each of these cases is equally likely as the other two. There is only one of these three cases where both pulled out green M&Ms. So, there is a 1/3 chance that given one pulled out a Green M&M, the other also did.

Conditional Probability

This type of probability where you say something like, ‘given X, what is the probability that Y’ is referred to as Conditional Probability. It is notoriously hard to understand, especially intuitively. We will be discussing conditional probability in future articles. One existing article which touches upon it is ‘A Whodunnit,’ a work in progress series about a murder.