Who gives a toss? The Solution.

If you toss a coin multiple times, what comes up first - a string of HTT or TTH? Or are they equally likely?

I love this puzzle because the answer is counter-intuitive.

For any individual toss of the coin H or T is equally likely.

And if I toss a coin three times, the result HTT and the result TTH are each equally likely. They both have a 1 in 8 chance of occurring.

But when I search for HTT and TTH in a string of results everything changes.

At the time of posting this solution, 80% of respondents to the poll said HTT and TTH were equally likely to appear first in a string of tosses.

Read on!

Write out a few lists of random results and inspect them for HTT or TTH. Then go back and check the first two tosses of every list. After a while you will realise that the only way to record TTH before HTT is by tossing TT to start.

Think about the first two tosses of any string of results. You can toss TT, HH, TH or HT.

If you toss TT first, you cannot toss HTT without having already tossed TTH. Don’t believe me? Try and write out a string of Ts and Hs that begin TT and get to HTT before TTH.

So if you toss TT to start, TTH will beat HTT.

If you toss HH to start, you should be able to see that you cannot get to TTH without having already encountered HTT on the list of results. Again try a few examples beginning HH and this will be clear.

So if you toss HH to start, HTT will beat TTH.

If you toss TH to start the same logic follows. At some stage you must toss TT and you have HTT before TTH.

So if you toss TH to start, HTT will beat TTH.

Finally, if you begin your tosses with HT, one of two things will happen. Your third toss might be a T. This gives you HTT. Or your third toss is a H. You then are in the same position as above. You cannot toss the TT needed for TTH without having already recorded HTT.

So if you toss HT to start, HTT will beat TTH.

In a string of coin tosses, HTT is three times more likely to occur before TTH.

How awesome is that!

This puzzle is actually known as Penney’s Game after its inventor, Walter Penney.

Penney gave us a simple rule to turn this mathematical exercise into a way to win money off your mates!

Explain the game to your friend. We are going to toss a coin and write down the results. We are trying to choose the triple that comes up first in the list of results.

Let your friend pick a triple.

Then you pick a triple.

If you play by Penney’s rule, over the course of a night you’ll win a lot more than you lose.

Let your friend pick first.

Whatever they pick, you take two steps. Drop their last coin from the triple. And put a coin in front so your first coin is different to your last.

Huh?

If they pick THT: drop the third coin so you have TH. Now pick a first coin different to the last one. Your last coin is H so pick T. Your response to THT is TTH.

The full table of choices for Penney’s Game is as follows:

No matter what they pick, if you follow Penney’s rule you can always pick a triple more likely to win.

If they pick THH, you pick TTH and win 2 of every 3 games you play.

If they pick TTT, you pick HTT and win 7 out of 8 times!

You won’t win every game, but over the night you’ll come out well ahead.

Happy hunting!