Probability puzzles
Puzzles that fool almost everyone, solved by counting carefully, including the famous Monty Hall problem.
Some probability questions look simple, and almost everyone gets them wrong. The most famous one is the Monty Hall problem, named after the host of an American TV game show. In 1990 a magazine writer, Marilyn vos Savant, gave the right answer in her column. Thousands of readers wrote to tell her she was wrong, and some of them had a doctorate, the highest university degree. She was right. The puzzles on this page do not need any new rules. They need careful counting, and you must be ready to check a feeling with numbers.
How to solve a puzzle
- Find all the possible results, and make sure they are equally likely. For two dice there are 36.
- Ask what exactly must happen. Write it in your own words.
- Count the results that work, or multiply the chances step by step.
- Check the answer another way: count the opposite, or use a shortcut like symmetry.
Symmetry means that two situations are mirror images of each other, so they must have the same chance. You will see it used below.
The Monty Hall problem
A prize is behind 1 of 3 doors. The other two doors hide nothing. You pick a door. The host knows where the prize is. He opens one of the other two doors, always one with no prize, and asks: do you want to switch to the last closed door?
Most people say: two doors are left, so it is 50-50, and switching does not matter. That is wrong. Your first pick was right with a chance of 1/3, and nothing the host does can change that, because he will always open an empty door. So the other closed door must hold the other 2/3.
| Where the prize is | You pick door 1. The host opens | If you stay | If you switch |
|---|---|---|---|
| Door 1 | door 2 or door 3 | You win | You lose |
| Door 2 | door 3 | You lose | You win |
| Door 3 | door 2 | You lose | You win |
The three rows are equally likely. Staying wins in 1 of them. Switching wins in 2 of them. So switching wins with a chance of 2/3.
Ties: a bigger number on the second roll
Example 1
You roll a die twice. What is the probability that the second number is bigger than the first?
- There are 36 results.
- In 6 of them the two numbers are the same (1-1, 2-2, ..., 6-6). These are ties.
- In the other 30, one number is bigger. By symmetry, the second is bigger in half of them: 15.
- 15/36 = 5/12.
Answer: 5/12
You can check by counting. If the first roll is 1, the second is bigger in 5 ways (2 to 6). If it is 2, in 4 ways. Then 3, 2, 1 and 0 ways. In total 5 + 4 + 3 + 2 + 1 + 0 = 15.
Odd and even
Many puzzles become easy when you only look at whether each number is odd (1, 3, 5) or even (2, 4, 6). On a die, each has a chance of 1/2. Remember these facts:
| First number | Second number | Product (multiply) | Total (add) |
|---|---|---|---|
| odd | odd | odd | even |
| odd | even | even | odd |
| even | odd | even | odd |
| even | even | even | even |
Each row has a chance of 1/2 × 1/2 = 1/4. Now read the answers from the table.
Example 2
You roll two dice and multiply the numbers. What is the probability that the answer is odd?
- A product is odd only when both numbers are odd (the first row).
- Both odd: 1/2 × 1/2 = 1/4.
Answer: 1/4
Example 3
You roll two dice and add the numbers. What is the probability that the total is even?
- A total is even when both numbers are odd, or both are even (the first and last rows).
- 1/4 + 1/4 = 1/2.
Answer: 1/2
This OR that: do not count twice
Example 4
You take one card from a deck of 52 cards (4 suits of 13). What is the probability that it is a heart or a king?
- There are 13 hearts.
- There are 4 kings, but one of them, the king of hearts, is already counted with the hearts. So add only 3 more.
- 13 + 3 = 16 cards: 16/52 = 4/13.
Answer: 4/13
All different, all the same
Example 5
Three friends each choose a number from 1 to 3 at random. What is the probability that all three choose different numbers?
- The first friend can choose anything.
- The second must choose one of the 2 numbers not taken: 2/3.
- The third must choose the last free number: 1/3.
- 2/3 × 1/3 = 2/9.
Answer: 2/9
Check by counting: there are 3 × 3 × 3 = 27 ways to choose, and the ways with all different numbers are the orders of 1, 2, 3, which is 3 × 2 × 1 = 6. And 6/27 = 2/9.
For "all the same" with a die, just multiply. Three rolls that are all 6: 1/6 × 1/6 × 1/6 = 1/216. There are 6 × 6 × 6 = 216 results, and only 6, 6, 6 works.
Try it yourself
1.You roll two dice and multiply the numbers. What is the probability that the answer is even?
Show the answer
3/4
The product is odd only when both are odd, 1/4. So even is 1 − 1/4 = 3/4.
2.You roll a die twice. What is the probability that the second number is the same as the first or bigger?
Show the answer
7/12
6 ties plus 15 results where the second is bigger: 21/36 = 7/12.
3.You take one card from a deck of 52. What is the probability that it is a club or a picture card (jack, queen or king)?
Show the answer
11/26
13 clubs, plus the 9 picture cards that are not clubs: 22 cards. 22/52 = 11/26.
4.Two friends each choose a number from 1 to 4 at random. What is the probability that they choose different numbers?
Show the answer
3/4
Whatever the first friend chooses, the second has 3 of the 4 numbers left that are different.
5.Three friends each choose a number from 1 to 4 at random. What is the probability that all three choose different numbers?
Show the answer
3/8
The second must avoid one number, 3/4. The third must avoid two, 2/4. 3/4 × 2/4 = 6/16 = 3/8.
6.A game has 4 doors and one prize. You pick a door. The host, who knows where the prize is, opens 2 other doors with no prize. You switch to the last closed door. What is your chance of winning?
Show the answer
3/4
Your first door wins only 1/4 of the time. In all the other cases, the prize is behind the one door the host left closed.
Ready to test yourself?
Play Probability: 20 seconds per question, and it gets harder as you go.