Gambler's ruin - problem 9

Probability Level 3

Notice that the wheel has 38 slots, 18 red, 18 black and two green. The wheel is spun with a ball in it. When it stops spinning, the ball comes to rest in one of the slots. All slots are equally likely.

The game works as follows:

  • You pay Cass a dollar, and declare a color, red or black.

  • Cass spins the wheel.

  • When the wheel stops spinning, if the ball falls into a slot of your color, Cass pays you two dollars. Otherwise you get nothing.

What is the expected profit to Cass?


The answer is 0.0526315789.

Varsha Dani by Varsha Dani , 47, USA

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2 solutions

Jordan Cahn
Feb 28, 2019

Cass loses a dollar when the ball lands on my color (18 slots) and gains a dollar when it does not (20 slots). Thus her expected profit is $ 1 × 20 38 − $ 1 × 18 38 = $ 2 38 ≈ $ 0.0526 \$1\times \frac{20}{38} - \$1\times\frac{18}{38} = \$\frac{2}{38} \approx \boxed{\$0.0526}

5 Helpful 0 Interesting 0 Brilliant 0 Confused
Gabriel Chacón
Mar 16, 2019

Another slightly different way to look at the question:

Cass will always get $1 at the start of the game but he will only have to pay $2 in 18 of 38 times on average. His expected profit is $ 1 − $ 2 × 18 38 ≈ $ 0.0526 \$1-\$2\times \frac{18}{38} \approx \boxed{\$0.0526}

1 Helpful 0 Interesting 2 Brilliant 0 Confused

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