# The Two Box Puzzle

### The Puzzle

Someone presents you with two opaque boxes. One has two $100 bills. The other has a $100 bill and a $1 bill.

Box | Box |
---|---|

$100 | $100 |

$1 | $1 |

You are told that you will get to keep the second bill you withdraw.

You pull the first bill, and it’s $100. Should you take the second bill from the same box or the other one—or does it make no difference?

### The Solution

There are two things you need to consider to answer this.

- The box you already drew from is down one $100 bill. The other box is guaranteed to have at least one $100 bill, whereas the box you drew from might not have one at all.

Now it might seem like it’s better to switch boxes. But there’s one more consideration to make:

- Before you pulled anything out, two of the three $100 bills were in the $200 box. When you pulled out a $100 bill, you got more information about the boxes: There is a 2 in 3 chance that the box you pulled from was the one with $200.

*If you have not learned about conditional probability, it might be difficult to wrap your head around point 2. This blog post gives a simple introduction to the concept.*

Let's put together everything we know.

**If you pull from the same box:**

- There is a 2/3 chance it is the box with two $100 bills. If it turns out this way, there is a 100% chance of getting $100.
- There is a 1/3 chance it is the box with both bills. If it turns out this way, there is a 0% chance of getting $100 since you already took out a $100 bill.

What are the odds of getting $100 if you pull from the same box?

2/3 x 100% + 1/3 x 0% = 2/3

There is a 2 in 3 chance of getting $100 if you pull from the same box.

**If you pull from the other box:**

- There is a 2/3 chance it is the box with one of each bill. If it turns out this way, there is a 50% chance of getting $100.
- There is a 1/3 chance it is the box with two $100 bills. If it turns out this way, there there is a 100% chance of getting $100.

What are the odds of getting $100 if you pull from the other box?

1/3 x 100% + 2/3 x 50% = 2/3

There is a 2 in 3 chance of winning if you pull from the other box.

Therefore, it doesn’t matter what you do!

## Courses

## Fallacy Detectors Part 1

Develop the skills to tackle logical fallacies through a series of 10 science-fiction videos with activities. **Recommended for ages 8 and up.**

## A Statistical Odyssey

Learn about common statistical misconceptions with an interactive space adventure. **Recommended for ages 12 and up.**

## Symbolic Logic for Teens Part 1

Learn how to make sense of complicated arguments with 14 video lessons and activities. **Recommended for ages 13 and up.**

## Emotional Intelligence

Learn to recognize, understand, and manage your emotions. Designed by child psychologist Ronald Crouch, Ph.D. **Recommended for ages 5 and up.**

## Worksheets

### Symbolic Logic Worksheets

Worksheets covering the basics of symbolic logic for children ages 12 and up.

### Elementary School Worksheets and Lesson Plans

These lesson plans and worksheets teach students in grades 2-5 about superstitions, different perspectives, facts and opinions, the false dilemma fallacy, and probability.

### Middle School Worksheets and Lesson Plans

These lesson plans and worksheets teach students in grades 5-8 about false memories, confirmation bias, Occam's razor, the strawman fallacy, and pareidolia.

### High School Worksheets and Lesson Plans

These lesson plans and worksheets teach students in grades 8-12 about critical thinking, the appeal to nature fallacy, correlation versus causation, the placebo effect, and weasel words.

### Statistical Shenanigans Worksheets and Lesson Plans

These lesson plans and worksheets teach students in grades 9 and up the statistical principles they need to analyze data rationally.