How would you bet?

Let's say that you own a biased coin that will land heads up with probability \( 1 > p > \frac{1}{2} \). You find someone who is willing you to offer you even odds that heads will turn up. They are also willing to bet as many times as you want.

Assume that you start off with $1000. Clearly, you would stand to gain a lot of money by betting on heads.
1. How would you maximize the final amount of money that you have?
2. If you want to reach $10000 as quickly as possible, how would you do that?

If you have no more money, then you cannot continue betting. E.g. If you bet $1000 in the first round, there is a possibility that you will end up broke and unable to continue.

Note by Calvin Lin
3 years, 4 months ago

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You stand to gain the most profit by betting the smallest possible amount that you can as many times as you can. This is because the Law of Large Numbers indicates that as the number of trials approaches infinity the recorded results will tend towards the expected probability.



The bigger your bet is, the more likely you are to go broke because your number of trials will be minimized. You should bet the smallest amount that the other person is willing to bet: pennies are better than dollars. If you bet a large amount of times, the rule is the following:

Let \(p[H]\) = The probability of flipping heads.

Let \($\) = The expected profit.

Let \(F\) = The number of flips.

\($ \approx 2F( p[H] - 0.5)\)

Here is some Python that uses random simulations to demonstrate this rule:

from random import random
bets = 10000000
heads_prob = 0.55
profit = 0.0
expected = 2 * bets * (heads_prob - 0.5)
bet_amount = 1
for bet in xrange(bets):
    if random() <= heads_prob:
        profit += bet_amount
        profit -= bet_amount
print "Expected profit:", expected
print "Profit:", profit

The Law of Large Numbers:

Brock Brown - 3 years, 4 months ago

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Unfortunately, by betting a small amount, that will take you a long time to accumulate any wealth. IE you do not want to be 80 and betting 1 cent, when you could be 40 and enjoying life from a yacht.

We we also want to reduce the number of coin flips that we have to do, what kind of strategy would make sense? Say that we can only flip the coin once a day. If \( p = 0.6 \), how quickly (minimum expected time) can we get from $1000 to $10, 000?

Calvin Lin Staff - 3 years, 4 months ago

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