I Am Not 6 Years Old, But I Can Solve This Probability Puzzle …

Barry Leung posted this puzzle today, asking, “Are you smarter than a 6-year-old?” Apparently, everybody is eager to prove they are smarter than a 6-year-old! Why is that?

And as usual, I am going to solve it with a Bayesian Network! It’s all I ever do!

This can be solved with the Geometric Distribution (Barry Leung’s image)

This gives us the number of failures before the first success.

Geometric Distribution as defined in Netica

First, let’s look at the expected number of throws it would take (much like how many flips of a fair coin before the first head).

This is Larry before and after compiling the Geometric equation

This says there’s a 50% chance he gets it on the first throw, a 25% chance it takes two throws, and so on. The expected number of throws is two. All we need now to calculate Larry’s chances of winning, given that he goes first, is to add a similar node for Julius and then compare them:

This game is VERY unfair!

Larry has hugely better chances than Julius, given that he goes first.

Can we make this a bit fairer? Suppose, instead, that Larry has really very bad aim, and his chances are only 1/3 to Julius’ chances of 50%. What are the chances Larry wins now?

This game is now fairer, Larry goes first to make up for his bad aim!

BUT can we make this game more interesting?

Here’s another way of modeling this question, instead of using the Geometric Distribution. Let’s model just one round. If they both miss, then they continue to the second round.

Notice that larry has a 2 to 1 advantage over Julius

Given Larry’s 2-to-1 advantage because he goes first, this translates to 2/3 chances of winning. We can recover the implied probability of winning by entering a negative finding, namely that the game does not continue.

The implied win probability for Larry when we eliminate ‘continue’

But again, if his aim is terrible, and his chances are only 1/3 vs 1/2 for Julius, then going first would make this game fairer. Let’s change Larry’s priors to 33.3% and see what happens:

Notice that if Larry’s chances are only 33.3%, then the game is fair and they have equal chances of winning

Another scenario:

Suppose instead that Julius can learn something by observing Larry, so that IF Larry misses, then Julius can improve his chances, say to 60%, just by observing what went wrong with Larry. That means that the nodes are no longer independent, and we connect them as follows:

This is a modest improvement in his chances

The implied win probability for Larry is Julius learns by observing Larry

These somewhat improved odds implies 62.5% chance for Larry winning.

SOURCES & REFERENCES

[embed]Are You Smarter Than A 6-Year-Old? *A Probability Puzzle medium.com*

[embed]Geometric distribution - Wikipedia In probability theory and statistics, the geometric distribution is either one of two discrete probability…en.wikipedia.org