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# 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 2 years, 9 months ago MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$...$$ or $...$ to ensure proper formatting. 2 \times 3 $$2 \times 3$$ 2^{34} $$2^{34}$$ a_{i-1} $$a_{i-1}$$ \frac{2}{3} $$\frac{2}{3}$$ \sqrt{2} $$\sqrt{2}$$ \sum_{i=1}^3 $$\sum_{i=1}^3$$ \sin \theta $$\sin \theta$$ \boxed{123} $$\boxed{123}$$ ## Comments Sort by: Top Newest 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. largenumbers 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:   1 2 3 4 5 6 7 8 9 10 11 12 13 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 else: profit -= bet_amount print "Expected profit:", expected print "Profit:", profit  The Law of Large Numbers: http://en.wikipedia.org/wiki/Lawoflarge_numbers - 2 years, 9 months ago Log in to reply 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?

Staff - 2 years, 9 months ago