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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?

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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?

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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 \(\frac{1}{2}.\)

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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.

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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?

**Definitions:**

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:

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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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