Math as an extension of common sense

I was leaving the house this morning and ran into a spectacle at the gate. A colourful fellow surrounded by the estate gatemen was telling them about a game they could play to win money. The economy being as it is, the words win and money go together like “Jollof” and “Wedding”.

It’s a simple game. A card holds the numbers 1 to 15 on one side and a hidden number and word combination you could scratch to see on the other. You picked a number and if it matched the number revealed you won. What you won was determined by looking up the revealed word on a table of prizes. Prizes ranged from ₦1,000 to ₦50,000.

You could pay ₦200 for the chance to pick one number, ₦1,000 for the chance to pick five and ₦3000 for the chance to pick 10. Immediately I could see that by offering the game to the gatemen for whom ₦1,000 was not a trivial sum he made sure the chance to win was always 1/15. Good odds for him.

On hearing this, he smiled with twinkle in his eyes and said “Oga make I do you small favour”. He offered to put up ₦800 of his own money to allow me reach the ₦1,000 for 5 numbers deal. I’d however have to pay him back his₦800 but only if I won. This seemed like a very good deal. Too good in fact. I had a hunch about how it worked so I took him up on it.

I picked 5 numbers, scratched and I won! ₦1000. After paying him his ₦800, my winnings were ₦200. Coincidentally the amount I put in. He smiled and said “Bros, your luck good. Try again, you fit win more”. He offered me the same deal, maybe this time I would win more than ₦1,000 and make a profit. I refused to take the deal and he tried his hardest to convince me. He couldn’t afford to look beaten in front of the onlooking gatemen, his soon to be customers.

I refused to play again here’s why. For one, he had two different sets of cards. Offering to put up ₦800 he took the card out of the front of his bag where I suspect all the prizes were ₦1000. When he did the sample for the gatemen he took out of the back and the fellow won ₦8000. Not to be paid of course as he didn’t put up any money.

The second reason was the odds. At this point we’ve established he didn’t stand to lose anything. As long as I kept winning ₦1,000 and paying him his ₦800 investment I merely broke even but I only had to lose once to be out ₦200. The only way to win was not to play in the first place.

The odds of my 5 out of 15 cards turning up is 5/15 or 1/3. but the odds of winning twice in a row is 1/3 x 1/3 or 1/9 which are much worse odds. By the third game I’d have a 1 in 27 chance of winning all three which is what I’d have to do just to take home my ₦200 investment. It’s a sort of reverse Monty Hall problem. By playing again I remove one choice and allow him a 50/50 chance of taking my money; by the third he had a 2/3.

I left with my ₦200 intact and a deeper appreciation of my mathematical education.

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