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Games of Chance

Some simple math can help you win games of chance. Learn how to be better than lucky with these fun puzzles.

Challenging Brain Tanglers


Try this quiz to challenge yourself with some difficult (perhaps even brain-warping) puzzles that will tie your thoughts up in knots!

Be forewarned that most people get these answers wrong, even when they think about it hard and carefully.

If you get a question wrong, compare your work to our solution to improve your own strategies!

A variant of the famous Monty Hall Game Show Puzzle:

You're a contestant of a game show! There are 10 closed doors: 9 lead to nothing and 1 leads to an expensive car. You are allowed to pick a door and earn the car if it's behind the door you choose.

Stage 1: You choose a door.

Stage 2: The host tells you to choose from two helpful options:

Option 1: Open Five doors!

You choose four more doors in addition to the one you've already selected and open all 5. You win the car if it is behind any of the five doors you choose and open.

Option 2: The host eliminates 8 red herrings!

The host will open 8 empty doors that are not the door you chose initially that do not contain the car. This leaves two doors closed: your initial choice and one other door -- the car is definitely behind one of them. You can then choose to either open the original door you chose in stage 1 or open the only other remaining closed door.

What should you do to maximize your chances of winning the car?

When I put two marbles in this bag, I flipped a coin twice to determine their colors. For each flip:

  • If it was Heads \(\rightarrow\) I put in a red marble.
  • If it was Tails \(\rightarrow\) I put in a blue marble.

You reach into my bag and randomly take out one of the two marbles. It is red. You put it back in. Then you reach into the bag again. What is the chance that this time, you pull out a blue marble?

When I put two marbles in this bag, I flipped a coin twice to determine their colors. For each flip:

  • If it was Heads \(\rightarrow\) I put in a red marble.
  • If it was Tails \(\rightarrow\) I put in a blue marble.

After I had the bag ready, I looked into the bag and announced, "at least one of marbles in this bag is red." To prove it, I took a red marble out of the bag and set it aside. I then asked you to reach into the bag and remove the only remaining marble. What is the chance that it is blue?

Hint: This the answer is not \(\frac{1}{2}.\)

In this game for two players, players take turns flipping a coin (they each have their own coin). Each round, if the flips match (HH or TT), they continue on. The game ends when the two flips in a round don't match (HT or TH). When the game ends:

  • Player 1 wins if it's HT.
  • Player 2 wins if it's TH.

An example game is pictured above. If this game is played with weighted coins that land heads 99% of the time and tails 1% of the time, which player would you rather be?

Note: "The game is fair," means that neither player has an advantage over the other.

For this puzzle, test your intuition by guessing without calculating, or do the calculation if you want.

Suppose you draw four cards at random from a 52 card deck, approximately how much more likely is a four-card flush, compared to four of a kind?


A four-card flush is four cards of the same suit (spades, hearts, diamonds, or clubs). For example:

Four of a kind is four cards of the same value (4 aces, 5s, jacks, etc.). For example:

Even if you got all 5 of the problems in this quiz wrong, don't fret. These were some tricky puzzles and they're exactly the kinds of situations that cause most people to slip up.

If want more practice and more to learn, check out the remainder of this Exploration! There are many quizzes that practice both simpler and more difficult variants of the techniques that have been called into play so far.

And there are many many more games to play and explore in depth, including some common casino games such as Blackjack and Craps. We'll help you master the skills needed for working with probability so that you can improve your strategy when playing these games and much more.


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