This tutorial is going to teach you about simulation. Specifically you will focus on:
You can use simulation to figure out complicated problems like this one.
Suppose a family has five children. What's the probability that at least two of those five are girls? You could figure this out in a tree diagram, but that would be a pretty unwieldy tree diagram.
One way to do it is to approximate the probability using a simulation. Get your dice out and let the boys and girls be equally likely, letting the faces one through three represent girls, and four through six represent boys. Use five dice.
Simulation/Monte Carlo Method
A way to approximate probability based on trials of chance experiments that mimic the real-life trials.
One through three represents girls, and four through six represents boys. The situation above has two girls in it. The one and the two.
Roll again.
This situation had four girls in it-- the two, the three, the three, and the one-- were four girls. Redo this many times and the relative frequency of getting two or more girls will approximate the probability.
Do this 20 times then analyze the results.
The first trial you got a 6, 4, 1, 2, and 6 and had two girls. See the rest of the dice rolls which are being tabulated behind the scenes. One situation didn't get any girls in your group of five.
This is the histogram of the number of girls per family.
Once you got 0 girls. It is pretty unusual that in none of the simulations did you get exactly one girl. You wouldn't expect that to happen again.
The most common was two girls, three and four was also very common. Don’t expect to get four girls as often as you did if you run this simulation again. In none of the families were all five of the children girls.
You got at least two girls in 19 of our 20 simulations. Your guess for the probability of having at least two girls in a family is 19 out of 20, or 95%. Your best guess is that you have a 95% probability of getting at least two girls in a family.
That is NOT the right answer. If you had done more simulations, you would have gotten a more correct answer because things would have started to even out. The right answer is in fact 0.8125, about 81% of the time.
If you simulated more than 20 times with the dice, you would have gotten a more accurate response.
This graph shows an approximation based on 10,000 simulations.
Your sample of 20 was not super representative. Apparently, four is fairly uncommon. Two and three are very common. In your sample of 10,000 families, 8,127 of the 10,000 had two girls or more. Your best guess now as to the probability that a family has two or more girls is 0.8127, which is a lot closer to the 0.8125 that the correct answer is.
It's possible to solve these complicated problems through a process called simulation. This is also called the Monte Carlo method, named after Monte Carlo Casino because it uses physical objects like dice.
First, come up with a way to simulate that has the same frequencies of what you're trying to predict. Here, we had boys and girls. Pick something that had a 1/2 probability, like three faces of the dice. Second, run many trials of your simulation. And third, answer the question based on the frequencies from your simulated results.
Good luck.
Source: This work adapted from Sophia Author Jonathan Osters.
A way to approximate probability based on trials of chance experiments that mimic the real-life trials.