Yesterday I offered you this Deal or No Deal situation (lifted from Monday's show) and asked what you'd do. [£5] [£50] [£100] [£750] [£5000] [£20000] [£35000] [£250000]
180 of you responded, thanks. Here are the results: Banker's Offer £3500: 7% deal, 93% no deal Banker's Offer £7000: 24% deal, 76% no deal Banker's Offer £17000: 66% deal, 34% no deal Banker's Offer £33000: 91% deal, 9% no deal
Only a handful of you would cave in and deal at £3500. A quarter of you would be willing to accept £7000. Two thirds of you would be happy to walk away with £17000. And almost all of you would accept £33000 and continue no further. By the looks of it, the knife-edge Banker's offer which would split you down the middle comes in at just over £10000. Which is very interesting. And very wrong of you.
Look at those eight amounts of money again. Two of them are very big but one, the quarter of a million, is absolutely huge. Numerically, the £250000 box dominates the list. It's worth more than four times as much as all the other boxes put together. Receiving a cheque for that amount of money would transform your life... and there's a one in eight chance that the £250000 is in your box. Those may not sound like great odds, but when before in your life has anybody ever offered you a one in eight chance of winning quarter of a million pounds. Nobody ever will again. Isn't it worth the risk?
Well maybe not. Every Deal or No Deal player has a certain notional amount of money they'd not be prepared to throw away. An amount which would make a big difference to their life and pay off a debt or three. An amount they'd not be prepared to gamble with, because the promise of 'cash-in-hand' is much stronger the risk of the unknown. Once the Banker's offer hits this magic amount then contestants cave in and stop. For some it's as little as £3500. For many (as we've seen) it's about £10000. Most people succumb once the offer rises into five figures. And that's great news for the Banker, because it means that (on average) people stop too early. Much too early.
Consider those boxes again. The eight amounts of money add up to a massive £310000. Divide that total by eight and your average expected winnings are approximately £39000. If the Banker was being perfectly fair, he should be offering you £39000 to stop. But he doesn't need to, because he knows that you have no concept of theoretical probability and you'll stop for less. Much less in fact. Most of you would stop for £17000, which is less than half what you deserve. Almost all of you would stop for £33000, and even that's below the statistical average. In reality, on Monday's Deal or No Deal, the Banker offered a paltry £7000 in this situation. And that's less than 20% of the expected winnings. Pitiful. Monday's contestant carried on, but according to my survey a quarter of you would have stopped, swindled.
And there's more to your delusion. Here again is the position the game is at: £5 £50 £100 £750 £5000 £20000 £35000 £250000 The next stage is to open three of these boxes, leaving just five. Sounds dead risky doesn't it? What if the big numbers are revealed? There's a 50-50 chance that the next round of box-opening will wipe out more reds than blues, and that would leave you in a worse position. Or so you'd think. You'd be wrong.
There are 56 different possible ways that the game could proceed. One would be to open £5, £50 and £100, which would be fantastic. Another would be to open £20000, £35000 and £250000, which would be a disaster. And there are 54 other possibilities inbetween. It's counter-intuitive, but you are nearly twice as likely to keep the £250000 box as you are to lose it. 64% of the time the big number stays. And if the £250000 stays, then your expected winnings (now averaged over just five boxes) must increase. Which means you can expect a better banker's offer after the next round two thirds of the time. You'd be a fool not to go on.
Except, as we know, chance is unfair. The Banker plays this game every day, so things average out. You only get to play ONCE. You can't necessarily afford to take risks. You're just as likely to have the five pounds in your box as the quarter of a million. Opening more boxes is a risky strategy, whereas a banker's offer is a dead cert. Keep going and that potential fortune could slip oh so easily through your fingers in seconds, because dreams can vanish if you push your luck too far. Most people placed in this situation are stubbornly risk-averse, and that's the reason why nobody on the programme has yet won the £250000. Six contestants have had the correct box in front of them, but none has been willing to progress far enough to open it. The show's producers are laughing all the way to the Bank. And we're still watching, because human fallibility makes the best television of all.
