Solving statistics question using Python

1
votes

I'm really lost. How can I approach this problem?

Beginning with $1 of capital, you choose a fixed proportion p of your capital to bet on a fair coin tossed repeatedly for 1000 times. Your returns is doubled if the toss lands head and you lose your it lands tail. For example, if p=0.25 and for the first toss your bet $0.25, and if heads appears you win $0.5, and so you have $1.50. You continue to bet $0.375 on the second attempt and if the second toss lands tail, you have $1.125 left.

Suppose p is chosen to maximise the chance of having at least a billion dollar after 1000 flips, what is the chance that you become a billionaire?

How can you use Python to code out this scenario and come out with an answer?

3
pythonstatistics
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3 Answers

1
votes

Do you have any experience with python? If not, read the tutorial.

To solve your problem, you should first write down some kind of pseudo-code. Your first attempt can be very general, and then you should go into more detail about specific operations, until in the end you actually go and implement it. Think about details such as, what pre-conditions do you have and what post-condition do you need?

1
votes

Here are some tips. The order of the wins and losses doesn't affect the total amount of money in the end because multiplications commute. Therefore, the total amount of money after all of the tosses (when starting with $1) is equal to 1 * (1+2*p)^(W) * (1-p)^(1000-W) where W is the number of total wins out of 1000 tosses (and therefore 1000 - W is the number of losses). This will allow you to determine if, for a given number of wins, W, you obtain more than a billion dollars. However, there are many more ways to win 500/lose 500 than to win 1000/lose 0. You can find the number of ways to have W wins out of 1000 tosses by using the binomial coefficient.

If you put these ideas to proper use then you can find a p that maximizes the probability. However, you should note that there is actually a range of p that all give an equal chance of going over a billion dollars. They do not all yield the same amount of money though.

0
votes

The obvious place to start is to write some code that will simulate performing 1000 coin tosses and give you a value for capital at the end. This is basically trivial:

def _mc(p):
    capital = 1.0
    for _ in xrange(1000):
        if random.random() < 0.5:
            capital *= 1 + p
        else:
            capital *= 1 - p
    return capital

Notice that capital will probably end up being tiny. That's fine.

Now this is obviously heavily dependent on what the random flips are, which is bad. So you should work out its expected value by performing lots of 1000-coin-flip-chains and doing some sort of statistics on what you think it should be.

Finally, you want do to all of this for a range of values of p, probably between 0 and 0.2. You could use matplotlib to plot a graph of p against expected outcome to get an idea of what p should be best.

Note that Python is probably not the best language for this sort of thing; C would be much faster and you don't really need the flexibility of Python anyway.