Gambler A chooses a series of three possible outcomes from successive throws of a die, depending simply on whether the number thrown each time is odd (O) or even (E).
Gambler B then chooses a different series of three successive possible outcomes. The die is then thrown as often as necessary until either gambler's chosen series of outcomes occurs.
For example, Gambler A might choose the series EOE and B might choose OEE. If successive throws gave, say, EEOOEOE, then A would win the game after the seventh throw. Had the sixth throw been E rather than O, then B would have won.
A has chosen the series EEE; and B, who was thinking of choosing OEE, changes his mind to OOO. Has B reduced his chance of winning the game, has he increased his chance of winning the game, or is it still the same? Provide sufficient reason for your assertion.
By changing to OOO, B has made the odds 50-50 as we can see the probability of each one is 1/2 by symmetry.
If B had stuck with OEE, he would have had a better chance of winning:
If the sequence had started with an O (1/2 probability) he would have been assured a win regardless of what happened next, as any series of three E's would have been preceded by an O and then the first two E's would have won for B.
But in addition to this 1/2 probability of winning, there would be other chances as well. In fact, the only way A could have won would have been if the first three throws were E, as even EO or EEO would have assured B of victory.
So if B had stuck with OEE, he would have had 7/8 probability of winning with A winning only on the 1/8 probable EEE at the very beginning.
Posted by Charlie
on 2013-04-24 12:22:59