Make it Simple
by Julie Brennan / June 2004
"The simplification of anything is always sensational." (G. K. Chesterton)
Want to understand something mathematical? Bring it down to simple numbers.
We were driving last night and my 12 y/o who has a decent grasp of probability ideas as they apply to games was ridiculing people who think they have increased chances of winning or losing on a dice roll as time goes by. He had grasped the idea that a penny has a 50/50 chance of landing heads or tails no matter how many times you flip it.
Hmm, I wonder how well he understands this? I ask him, "Troy, who has a better chance of winning. A person who buys one lottery ticket every week for a month, or a person who buys four tickets once a month?"
8 y/o brother DJ in the back yells "The person who buys a ticket every week!"
I ask, "Why? What are the chances of winning?"
Troy: " Uh, one in millions?"
I say, "So, what are the odds of each ticket winning? - In the first case, each ticket has, say, a 1/1,000,000 chance. In the second case, the first ticket has the same chance, but the 2nd has a 1/999,999 chance, and the 3rd 1/999,998 chance, and so on, do you see that?"
Both boys are having a hard time with the big numbers.
I say, "Okay, let's say you are buying raffle tickets, and there are 100 tickets for each drawing. If you buy 1 ticket for each of 4 drawings, or 4 tickets for one drawing, which would give you a better chance of winning? What are the odds in each case?"
Troy can figure that out: "The first case he has a 1% chance of winning every time. In the 2nd case, he has a 4% chance." Boys are STILL having trouble grasping however, that you can't ADD the 1% cases together to get 4%. They still can't shake the luck aspect and think more drawings increase their chances.
"Okay, let's put it in other terms. Both people have $100 dollars to spend on the dollar raffles. Who has the better chance of winning, the person who spends $100 on one raffle, or the person who spends $25 on each of four raffles?"
Troy gets it now - "The first guy." DJ still doesn't get it though, "Why?" Troy: "Because he has a 100% chance of winning, and the other guy only has a 25% chance of winning each of the other raffles." 8 y/o - "But the other guy could win FOUR times!" Troy's the teacher now: "But is he LIKELY to?" DJ: "Hmm, no, not with only a 1 out of 4 chance of winning, he's probably not going to win anything unless he's lucky . . ."
So I go on to talk about the aspect of people who buy a weekly $5 lottery ticket, vs. pooling their $20 a month with others to purchase group tickets, and why this increases the chances of winning, even if they have to share the winnings.
DJ, the practical one always, insists: "But someone may spend $100 to win a prize that isn't worth that!" I say, "Undoubtedly that is true, why would a fundraiser give away prizes worth more than the money they take in? But people make foolish decisions all the time when they don't understand value and chances." And the conversation moves on into some interesting economics ;o).
Make it simple. Converting big numbers to little numbers through rounding and estimating is the only way we can comprehend large number situations. We have no real concept of 15,000,000. But 25 out of a 100 - most children understand the relative value of a quarter to a dollar.
June 3, 2004