Man most feared by Casinos

Probability Level 3

In most casino games, the odds are approximately 51:49 (or better) in favor of the house. Suppose 4 players A, B, C, D each with $10,000 play one of these games with 51% chance of losing their bet and 49% chance of winning the equivalent amount. The following are their strategies:

A: Bet $2,500 each across 4 rounds.
B: Bet $5,000 each across 2 rounds.
C: Bet half in Round 1. Bet half the balance from Round 1 in Round 2.
D: Bet all $10,000 in Round 1.

From the standpoint of a player, the higher the expected return (or equivalently, the less the expected loss), the better; the higher the probability of staying alive (with positive balance) after completing their strategy, the better.

Which player's strategy is absolutely superior over all the other strategies, with respect to both expected return and probability of survival?

In other words, who would be the player most feared by casinos?

Bonus: Try more unfair odds like 60:40 or 70:30 for a deeper insight into this problem.

A B C D No strategy is absolutely superior

Jimin Khim by Jimin Khim

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1 solution

Jimin Khim Staff
Sep 29, 2017

First, let's compare only A, B, D:

  • EXPECTED RETURN
    Let x x be a bet. Then, even though the actual return on the bet will be either − x -x or x , x, the expected return on the bet (before the bet x x is played) is ( Probability of Winning ) × x − ( Probability of Losing ) × x = 0.49 × x − 0.51 × x = − 0.02 × x . (\text{Probability of Winning}) \times x - (\text{Probability of Losing}) \times x = 0.49\times x-0.51\times x={\color{#D61F06}-0.02}\times x. (This means they basically give away a small portion of their bet every time, on average.) So, for A, B, D, the expected returns on their strategies are

    • A: 4 × ( − 0.02 × $ 2 , 500 ) = − $ 200 4\times (-0.02 \times \$2,500) =-\$200
    • B: 2 × ( − 0.02 × $ 5 , 000 ) = − $ 200 2\times (-0.02 \times \$5,000) =-\$200
    • D: 1 × ( − 0.02 × $ 10 , 000 ) = − $ 200 , 1 \times (-0.02 \times \$10,000)=-\$200,
      which are all equal. \hspace{30cm}

  • PROBABILITY of SURVIVAL
    Since each of the players A, B, D stays alive unless each loses all their rounds, their probabilities of survival are

    • A: 1 − 0.5 1 4 ≈ 0.93 = 93 % 1- 0.51^4 \approx 0.93=93\%
    • B: 1 − 0.5 1 2 ≈ 0.74 = 74 % 1- 0.51^2 \approx 0.74=74\%
    • D: 1 − 0.51 = 0.49 = 49 % , 1- 0.51 = 0.49=49\%,
      of which A is the greatest.

Therefore, without C , player A 's strategy is relatively superior, with expected return -$200 (tied) and probability of survival 93% (greatest).


Now, since C always leaves half its current balance behind for coming rounds, the probability of survival is by definition 100%, which is superior over A 's 93%. ( 1 ) \qquad (1)

Let's see if the same is true of expected return:

  1. Win-Win (Balance Flow $10,000 → \to $15,000 → \to $ 22,500) with probability 0.49 × 0.49 = 0.2401 : 0.49 \times 0.49 = 0.2401: Expected Return = ( $ 22 , 500 − $ 10 , 000 ) × 0.2401 = $ 3001.25. \text{Expected Return} = (\$22,500-\$10,000) \times 0.2401 = \$3001.25.
  2. Win-Lose (Balance Flow $10,000 → \to $15,000 → \to $7,500) with probability 0.49 × 0.51 = 0.2499 : 0.49 \times 0.51 = 0.2499: Expected Return = ( $ 7 , 500 − $ 10 , 000 ) × 0.2499 = − $ 624.75. \text{Expected Return} = (\$7,500-\$10,000) \times 0.2499 = -\$624.75.
  3. Lose-Win (Balance Flow $10,000 → \to $5,000 → \to $7,500) with probability 0.51 × 0.49 = 0.2499 : 0.51 \times 0.49 = 0.2499: Expected Return = ( $ 7 , 500 − $ 10 , 000 ) × 0.2499 = − $ 624.75. \text{Expected Return} = (\$7,500-\$10,000) \times 0.2499 = -\$624.75.
  4. Lose-Lose (Balance Flow $10,000 → \to $5,000 → \to $2,500) with probability 0.51 × 0.51 = 0.2601 : 0.51 \times 0.51 = 0.2601: Expected Return = ( $ 2 , 500 − $ 10 , 000 ) × 0.2601 = − $ 1950.75. \text{Expected Return} = (\$2,500-\$10,000) \times 0.2601 = -\$1950.75.

Overall, the expected return on C 's strategy is $ 3001.25 − $ 624.75 − $ 624.75 − $ 1950.75 = − $ 199 > − $ 200 , \$3001.25-\$624.75-\$624.75-\$1950.75=-\$199 >-\$200, which proves C 's superiority over A in expected return. ( 2 ) \qquad (2)


Therefore, from (1) and (2), C is absolutely superior over all the other players, with respect to both expected return and probability of survival. □ _\square


One very important lesson to be learned here is, "Get aggressive (raise your bet) when you are successful, and stay conservative (reduce your bet) when you are not doing well." But the majority of people do exactly the opposite--which is an unavoidable part of human nature--and leave casinos early, broke.

Try using more unfair odds like 60:40 or 70:30, and you will see the importance of appropriate bet size adjustment more vividly.

Remember, gambling is more about self-control than probability--in reality, if not in theory. Knowing this, casinos fear those who are scarily good at self-control, not those who are good at math!

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