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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.

This Exploration helps you learn how to **optimize your chances to win probabilistic games**, like Blackjack and Craps!

In this first quiz, we'll cover **5 quick "pro tips"** for game strategy.

Whether you're a professional gamer or a newbie to most gaming, you'll have fun with these puzzles - and the scenarios get trickier as this quiz progresses.

**Pro Tip 1: Don't be overly tempted by large but unlikely payoffs.**

Especially if you're playing a game many times, the best strategy often builds up gains slowly, rather than making a risky gamble for a large payoff.

You are about to play a game with a coin that is **weighted so that there is a 90% chance that it lands heads and a 10% chance that it lands tails.** If you want to maximize the expected amount of money you will win, would you rather...

**A.** win **$10** if it lands **heads,**

**B.** win **$50** if it lands **tails,** or

**C.** win **$1,000,000** if it **spontaneously combusts in the air.** (You may assume that there is at most a 0.0000001% chance that the coin will spontaneously combust in the air.)

Choose wisely!

**Pro Tip 2: Look for a "simplifying perspective" when there are many possibilities.**

In the problem below, you can map out all of the cases in a few minutes, but there's also a very clever, quick way to solve it using the right perspective!

Four mice are on the corners of a square. All at the same time, they each randomly and independently choose to walk along one edge of the square to a new corner. **What is the probability that at least two of the mice collide before reaching their destinations?**

**Pro Tip 3: Your intuition may mislead you.**

For example, the "gambler's fallacy" is when a gambler believes that they're more likely to win than usual because they've been losing for so long that a win is "due."

You are going to flip 8 fair coins in total, and three of your first four flips have already landed tails. **What do you expect the total count of tails to be when you're finished flipping all 8 coins?**

**Pro Tip 4: When you're studying a complex game, simplify it by breaking it into smaller games and/or scenarios that you've already worked with before.**

For example, drawing a black card vs. a red card from a deck is just like flipping a coin if all you care about is the card's color, and \(\frac{1}{2}\) is a much easier fraction to mentally keep track of than \(\frac{26}{52}\).

Katherine and Zyan are playing a game using strange dice. Each die is a cube with six sides. Katherine's die has sides numbered 3, 3, 3, 3, 3, and 6. Zyan's die has sides numbered 2, 2, 2, 5, 5, and 5.

To play the game, Katherine and Zyan roll their dice at the same time and whoever rolls the higher value wins. **If they play many times, who is likely to win more frequently: Katherine or Zyan?**

**Pro Tip 5: "Most likely" does not necessarily mean "very likely."**

When you're playing a game like poker where you can't know everything about the cards in play, don't play as if the most likely outcome is the only thing that could happen. Keep in mind that even if something is **most likely** it may still be very unlikely if there are lots of possibilities.

In Pokémon Go, eggs of this type have a **50%** chance of hatching an Onix.

Which is more likely?

A. Hatching **2 such eggs** and getting **exactly 1** Onix.

B. Hatching **2000 such eggs** and getting **exactly 1000** Onix.

**There's no need to calculate out these values exactly (one of these two probabilities is far larger than the other)--what's your intuition?**

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