Saturday, 27 September 2014

Million Pound Drop

I'm really rather keen on this quiz.

For anyone that doesn't know it, the contestants start with £1 million in bundles of £50K. They have to answer 8 questions, placing the money they have on 4 possible answers. If you're absolutely certain of a question, you stick a million on. If you can't decide between 2 possibles, you maybe put half a million on each. That then means you have half a million for the next question and so on. The last question is simply that all the money goes on your answer (out of 2).

The reason they aren't constantly handing out huge amounts of money is that most people aren't very good at compounding. They'll sometimes get an answer they're certain of, but stick a little on another answer, just as a backup in case, but of course, each time you do that, you reduce the stake.  So, if you start with £1m and place 1/8th of that on another answer, even if you're lucky enough to be quite certain, here's where your money goes:-

After Q1: 875,000
After Q2: 750,000
After Q3: 650,000
After Q4: 575,000
After Q5: 500,000
After Q6: 425,000
After Q7: 375,000

(I've stuck with rounding to the nearest £25K as the game requires that). So, even if you get some certainties and put a bit on "but it might be", you lose most of your money. In reality, there's quite a lot of answers which are 1 of 2, and of course, 3 of those and you've lost 7/8ths of your cash. And on the odd occassion that someone does get the wrong answer and has 125,000 after the first question, they never make it through with anything, because the remaining questions will whittle it down long before the end.

So, typically, people either end up with nothing, or maybe £25K, and a lot of it is because they're overcautious. If people played with calculated risks "I'm really certain of this answer, so we put the lot on it" they'd have better outcomes.

Pablo said...

You know this one? There are 3 boxes: one of them contains £1M. You have to pick one. Of the 2 remaining boxes an empty one is discarded. You are asked if you want to stick with the one you picked, or swap it for the other box? What do you do?

Bayard said...

This is the famous Monty Hall problem and it can be demonstrated that it is always better to switch.

Rich Tee said...

The key point about the Monty Hall problem is that the host knows what is behind the doors and opens one of the doors for you after you have already chosen one.

This is a really important point.

This is far more fun that the Monty Hall problem, but I've never watched MPD.

Is the rule that you can carry forward all the money you staked on the right answer in each round? Is that a fair summary?

The Stigler said...

Mark,

Yes, that's right. And the answers are often not facts like "who won the battle of Britain", but more like stuff you don't know the answer to, but two options seem most likely. So, the money gets split quite quickly.

It's a good game/quiz show.

Bayard said...

RT, it's just straight probability theory, thus:

At the beginning, you have three choices, two empties and a £1M, so your chance of picking £1M is 1/3. However, later you are asked whether you want to switch, so there are now six possible outcomes (two of which are the same):

Choose £1M, switch and get nothing
Choose £1M, don't switch and get £1M
Choose an empty, switch and get £1M
Choose an empty, don't switch and get nothing,
Choose an empty, switch and get nothing
Choose an empty, don't switch and get nothing,(there are two empties)

Each has a probability of 1/6.  However, outcome no 5 is ruled out by discarding one of the boxes with nothing in it.  Outcomes 4 and 6 are the same, so they can be combined with a probability of 1/3. Outcome 2, the only other outcome that involves not
switching, has a probability of 1/6, so if you don't switch, it's twice as likely you will pick an empty box as it is that you will pick one with £1M in it.