In my office, we have a small toy M&M vending machine that dispenses about a handful of M&Ms at a time. My coworker began a game of asking me if there at least 2 M&Ms of the same color out of 4 M&Ms dispensed. So he carefully gets 4 M&Ms from the toy vending machine. The machine holds probably about 200 M&Ms (not sure if that's important) that we regularly refill. The source of M&Ms has 6 colors. Our initial guess is that there's a 50% chance of a match. (3 chances that the color matches the first drawn M&M, so 3 out of 6 for 50%).
If we assume that the colors are in equal proportions (which they are not), are we correct in assuming the 50% probability?
$$\frac{3}{6}=0.5$$
On the other hand, if we break down each probability:
- First Draw: Not considered.
- Second Draw: 1 out of 6 probability of matching the first draw.
- Third Draw: 2 out of 6 probability of matching the first or second draw.
- Fourth Draw: 3 out of 6 probability of matching the first, second, or third draw. $$\frac{1}{6}*\frac{2}{6}*\frac{3}{6}=0.0278$$
But that seems wrong.
We are not mathematicians or statistician in this office btw. We are writers and researchers - so please go easy on me! We tried to ask one of our engineers, but he got obsessed with the design of the toy vending machine.
My question seems similar to this one, but not sure if that got answered: Probability of matching faces on 2 separate pools of d10s.