Bayes' Theorem

Probability and Statistics: Part 10

By Madison Nef

Conditional Probability

Conditional probability comes up in many different areas of probability and statistics. To explain it simply, imagine that you have 27 cards. You are then asked the probability of choosing a face card from the 27 cards given. 12 out of these 27 cards are face cards, so we know that the probability of selecting a face card from the cards given to us is 12/27. We are then asked the probability of choosing a red card out of the 27 cards. There are 6 red cards. So we know that the probability of choosing a red card is 6/27.

However, we are THEN asked the probability of choosing a red card, if we have already chosen a face card… and this is what conditional probability is. Only 3 out of the total number of red cards are also face cards… and as we know from before, there are 12 face cards that can be chosen out of 27. To get the probability of choosing a red card that is also a face card, we have to multiply the two probabilities. Since we have 12/27 probability of getting a face card, and OUT of those 12 cards we have a 3/12 chance of getting a red card… after we multiply the two fractions, we are left with a 1/9 chance of getting a red card after having already picked a face card.

 This method can also be reversed and used to determine the probability of choosing a face card when you already know you have a red card. You know that 6 out of the total 27 cards are red, giving you your first fraction:

6/27

You then find that out of the 6 red cards, only 3 of them are face cards. This gives us our second fraction, 3/6 (or ½ simplified), so:

6/27 * 3/6 = 1/9 (simplified)

While these two examples of conditional probability are very similar, they are actually of two different types, simply because of their reversal. This can be done to any conditional probability question… while it will indefinitely yield the same answer as the reversal, the two are in no way the same question OR the same conditional probability.

Bayes’ Theorem

The examples given above are an example of Bayes’ Theorem. The theorem simply states that the probability of B multiplied by the probability of A given B would always equal the probability of A multiplied by the probability of B given A. To simplify it, think of A as equaling red cards and B as equaling face cards, as written below:

P[B] x P[A/B] = P[A] x P[B/A]

A = red cards

B = face cards