Today I dropped my hand into a bag of m&m's and found that, without looking, I had pulled out 6 orange ones and one yellow one. This struck me as pretty unusual, so I started wondering what the exact chances would be. It's been a while since I mathed regularly, though, and probability was never my strong point. My rough guess, backed up by some rudimentary modeling in Python, is "really unlikely, but probably more likely than drawing six out of six identically colored m&m's". That's not a very interesting answer, though, so I was hoping someone here could help me out.
Here's the problem more precisely stated: given a bag of m&m's with equal numbers of each of the six colors (red, yellow, green, blue, brown, and orange), what is the likelihood that a randomly chosen handful of seven will contain at least six of the same color? We may assume that the bag is infinite if that helps simplify things; I doubt it would make a noticeable difference except for those dumb "fun-sized" bags (if I'm mistaken here, though, please point it out!)
Bonus points for:
- Showing the probability that exactly six out of seven will be of the same color
- Comparing to the probability of drawing six out of six identical m&m's (no "wild" m&m)
- Generalizing for arbitrary numbers of colors, m&m's drawn, and "wild" (extra) m&m's