Imagine you had an ‘unfair’ coin that returned heads 70% of the time and tails 30% of the time and you played this game with a friend (who knows that the coin is ‘bent’):
You each put £1 in as stake money every round and then toss the coin four times.
If it comes up as four heads, you take all the money in the pot, if it comes up as two heads and two tails (in any order), your friend takes the pot. If neither comes up, you play another round putting another £1 each in the pot.
Who – if you play for long enough – has the chances of coming away with more money?
The answer – and this is what seems strange to me – is your friend – as the odds of exactly two heads and two tails are 0.2642 compared to 0.2401 for exactly four heads.
So even though on each flick the chances of the coin turning up a head are more than twice those of it turning up a tail, the odds that half of the tosses result in tails are better than them all ending in heads.
(If you played it 100 times then your friend could expect to have made a profit of £4.85)
cartesian product ... stuff about computing mostly 