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Claiming an Edge


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

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.

In European Roulette, bets are made on a table like the one shown below and a wheel is spun with a ball that lands between the numbers 0 - 36 (inclusive). After the ball lands all bets are paid off or lost.

One bet that pays equal odds is the "Odd" square bet. If you bet some amount of money on "Odd," and an odd number comes up on the roulette wheel, the payout is twice your original bet. If you lose, you simply lose the money that you bet. Is this a fair bet (that is, are you and the casino equally likely to make money)?

Let's test your intuition!

Suppose you draw four cards at random from a 52 card deck as depicted below. Four cards of the same suit (a partial flush in spades, hearts, diamonds, or clubs)

is more likely than four cards of the same value (four of a kind: 4 aces, 5s, jacks, etc.)

But approximately how much more likely is a partial four-card flush, compared to four of a kind?

Just guess and test your intuition to choose an answer, or do the calculation if you want. (There is actually a fast shortcut for fully solving this quickly.)

Suppose you're playing a dice game against an opponent who is rolling two dice, whereas you are only rolling one die.

Every time he rolls a sum of 7 on his two dice, he gives you a dollar. Every time you roll a 6 on your single die, you give him a dollar. If you both roll an equal number of times, is this a fair game (are you both equally likely to make money)?

Suppose you draw cards at random from a 52 card deck, and you earn or lose money based on suits.

Is this a fair game (that is, on average, do you gain as much money as you lose)?


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