A bet on \"black\" in Roulette has a probability of 18/38 of winning. If you win
ID: 3266893 • Letter: A
Question
A bet on "black" in Roulette has a probability of 18/38 of winning. If you win, you double your money. You can bet anywhere from $1 to $100 on each spin. a. Suppose you have $10, and are going to play until you go broke or have $20. What is your best strategy for playing? Explain using information you learned in this module's material. b. Suppose you have $10, and are going to play until you go broke or have $30. What is your best strategy for playing? Explain using information you learned in this module's material.Explanation / Answer
A) The best strategy is to bet everything. This lets you win with probability 18/38, and you go broke immediately if it doesn't work. It is relatively easy to prove that this is optimal.
Let X be the amount of money you have at the end. E[X] = $20 * P(win).
This is also the initial amount minus your expected loss, which is the house advantage fo 1/19 times the amount you wager.
To reach $0 or $20, you have to wager a total of at least $10, so the maximum expected value you can have at the end is 10 - 10/19 = $10(1-1/19).
So, 20P(win)<=10(11/19),P(win)1/2(11/19)=18/38.
That means you can't win more than 18/38 of the time, so betting $10 on one color is optimal.
B) The bold strategy is to bet $10 when you have $10, and bet $10 when you have $20 since you only need to reach $20. What is your probability of winning?
We can set up a system of equations for this.
Let p be the probability that you win from $10, and let q be the probability that you win from $20.
P = 18/38 q since you need to win your first bet, and then you have $20
if it works q = 18/38 + (20/38)p.
You can use the first equation to substitute in the second.
qq(20/38)(18/38)q
q-(20/38)(18/38)q=18/38
q(1(18/38)(20/38))=18/38
q=(18/38)Divided by /1-(18/38)(20/38)
171/271=0.630996/ 81/271=0.298893
This is the same as if you use the following strategy: Aim for $15 by betting $5 at a time. If you reach $15, then bet $15 to double up or go broke. If you try this strategy, then you might bounce back and forth between $5 and $10 for some time. If you let p be the probability that you reach $15 from $5, and let q be the probability that you reach $15 from $5, these satisfy exactly the same equations as before! So, the probability of reaching $15 from $10 when you bet $5 at a time is q=171/271.
To reach $30 you need to succeed when you bet $15, which happens with probability 18/38, so the chance to succeed is (18/38)(171/271) = 81/271 = 0.298893.
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