Breakfast Math

Sometimes, marketing opens the door to a great educational experience.

Our box of Raisin Bran proclaimed, in large letters, "$100 CASH CARD INSIDE". In smaller letters, one realizes that in truth, "you could find up to a..." If you read the fine print, you find that your odds of finding that $100 cash card are 1 in 24,800. There are other prizes, though: a $50 cash card (1 in 20,667), a $25 cash card (1 in 17,714), a $10 cash card (1 in 15,500), and a $5 cash card (1 in 12,400). So I read these statistics to the kids and solicited guesses on just how many boxes of Raisin Bran one would have to buy to be likely to get any cash card at all. Then I hauled out my calculator and did the math: the total odds, obtained by adding the odds of each value of cash card, are 1 in 3444. If you bought 3444 boxes, though, you still wouldn't be guaranteed to get a cash card (though you'd have better than even odds). We discussed how long it might take to eat 3444 boxes of cereal (about a decade, if the whole family only ate Raisin Bran for breakfast every single day - we could probably polish off a box a day between us). And if you bought 3444 boxes of cereal, you'd most likely only get one cash card, which would most likely be a $5 cash card.

P and E - mostly E - have invented a useful number, the gi (hard g, rhymes with pea). One gi is generally defined as "a number larger than the one you were just talking about by a considerable margin". So we find it helpful to suggest, "even if you tried gi times, you wouldn't be able to throw an apple all the way up to the moon," or "do I have to ask you gi times to clean your room?" E announced, "If you bought a gi of cereal boxes, you'd probably only find 1000 cash cards. That's how big a gi is."

Okay, I'm curious. How large does gi have to be in order for 1000 to be the most likely number of cash cards when the odds of finding one is 1 in 3444? I think all the probability and statistics I was taught at Caltech has fallen out of my head. Anyone care to remind me?