Conditional Probability

Solving tricky probability puzzles like the Two Box Puzzle, the Two Child Problem, and the famous Monty Hall problem requires an understanding of conditional probability. The key to understanding conditional probability (and these puzzles) is realizing that we need to update our probability judgments when we get new information.

Let me explain with three examples.

The Strange Library

Imagine you walk blindfolded into a library in which half of the books are science-fiction novels and half the books are romance novels. The books are scattered randomly throughout the shelves. You pick a book.

What are the odds that the book you’re holding is a science-fiction novel? 1 in 2, or 50%.

Now, a friend walks into the library and says, “I like the spaceship on the cover of the book you’re holding!” You now know it is more likely that you are holding a science-fiction book.

The book you’re holding didn’t change; what changed is the information you have. You need to adjust your probability judgment accordingly.

The Four Balls

Imagine two people, Kate and Greg, are sitting in a room with two boxes, Box A and Box B.

The boxes are transparent on one side. Kate is sitting on the transparent side, so she can see what’s in the boxes. Greg is sitting on the other side, so he cannot.

Each box contains two balls. One box has two red balls, the other has one red ball and one green ball.

Box	Box
red	red
red	green

If Greg picks one ball out of a box at random, what are the odds that he will pick a green one?

1 in 4.

Greg then decides that he will pick a ball out of Box B and he tells Kate his decision.

What are the odds that Greg will pick a green ball?

From Greg’s point of view, the odds are still 1 in 4.

However, Kate can see what’s in Box B. She sees that it has one ball of each color. From her point of view, the probability that Greg will withdraw the green ball is 1 in 2.

1/4 and 1/2 are both correct answers, depending on your point of view.

The Money Boxes

Greg is alone in a room with two opaque boxes. One of them contains ninety-nine $100 bills and one $1 bill. The other box contains ninety-nine $1 bills and one $100 bill.

Box	Box
99 x $100	99 x $1
1 x $1	1 x $100

If he were offered to take one box home, he would have no reason to pick one over the other. At this point, the odds of either box being the Big Money box are 50/50.

Before making his decision, Greg is permitted to remove one bill. He does so, and it’s $100!

Which box should he choose to take home now?

It’s highly likely that his $100 bill came from the box with ninety-nine $100 bills, so this is the box he should take home. Had he picked out a $1 bill, it would have been better to switch.

These examples demonstrate that when we get new information, we need to update our probability judgments accordingly—even if nothing else changes.