A couple of weeks ago Nassim Taleb posted the following probability puzzle on Twitter.
— NassimNicholasTaleb (@nntaleb) November 2, 2017
I love puzzles so I took a stab at this using two different approaches. You can download my Excel spreadsheet solution here.
Approach 1: Probability Theory
I recognized the problem as being an example of binomial probability which I used previously in the birthday problem. We want to know the probability of having x successes from N trials where the probability Π of success in any one trial is constant. It’s a little trickier because there are 16 squares on the grid, so to begin I simplify it by calculating the probability of getting three darts in one specific square.
Note: It wasn’t clear to me whether we should calculate for exactly three darts in a square or at least three darts in a square. I did both as the extra computation is trivial.
x = 3 ; N = 8 ; Π = 1/16 = 0.0625 – plug these numbers into the following formula and you should get 0.0099, that is the probability of getting exactly 3 darts in a specific square, e.g. the top left square.
To compute the probability of getting exactly three darts in any square, simply multiply by 16 and you get 0.1584. Now to extend this and compute the probability of getting at least three darts in any square, simply repeat the above for x = 4, 5, 6, 7, 8 and sum the answers together. Or sum the probabilities for x = 0, 1, 2 and subtract from 1. Either way you should get an answer of 0.1723, that is a 17% chance of getting at least three darts in any square.
Approach 2: Simulation
Let’s say you understand the problem well but you’re not so hot at probability (that’s ok by the way, probability is hard, I’ve regularly seen experts make mistakes!) BUT you can code. If you can code you can randomly simulate the problem over and over again and just see how the results pan out. Heck, you don’t even need code per se, I did this in Excel – download my spreadsheet here. The simulated results matched the theory pretty well – 0.1542 and 0.1699 for exactly 3 darts and at least 3 darts respectively.
Doing fun puzzles like this is a great way to stay sharp statistically, think of it like a work out in the gym. And even if you don’t solve the puzzle, trying to solve it before peeking at the solution is how you expand your statistical knowledge. I recommend looking at all the replies to the initial tweet to see how some people got it wrong and how others got it right. And remember there can be multiple possible solution methods. I’ve spoken previously about the importance of tackling problems in more than one way in the Monty Hall Problem. It’s good to have these different approaches in your arsenal because:
- You never know when you’ll need them.
- Our knowledge is solidified when we understand a problem from different angles.
- If we’re not 100% confident in our answer, another approach can serve as a validation.
- Depending on the audience you may have to try different explanations to ensure they understand.
PS: Things can get a little funky when you start considering the cases where there are three darts in more than one square. I ignored that here but it does explain the slight difference between the theoretical solution and the simulated solution.