How to size bets
February 2022

An interactive article that discusses the optimal strategy for the following scenario:

I have a coin that lands heads 60% of the time and tails 40% of the time. You have $25 and can bet on either side of the coin — every time you’re right you double your bet, and every time you are wrong you lose it.”

Theres a widget on the page that lets you select an amount, what side to bet on, and a chart showing your balance after each coin toss.

My first attempt at this game was probably my best. My intuition is that its always optimal to bet on heads because each coin toss is independent, and 60% is more than 40%.

However the question of how much to bet on each coin toss is more complicated because you lose if your balance dips below 0. If you bet 50% of your balance and lose on 2 consecutive coin tosses, then you’re out. The probability of this is \(0.4^2 = 0.16\). 0.16 is quite low, but whilst playing the game you will toss the coin 50 - 100 times, so there are many opportunities.

If you toss the coin 50 times, there are 49 opportunities to lose twice consecutively. I don’t know the math for this probability, so I’ll go learn it, but intuitively I think that 50% of the pot is too high. I went with 33% and won (reaching \$250) in about 50 throws.

I’d like to know how to optimise this, so I’ll read the article.