For those who are unaware (since I’m not sure how regional this is), a game of odds or evens works like this: a player calls “odd”, the other calls “even”, and then they simultaneously show each other their hands – usually just one hand per player – with a number of fingers held up (or no fingers, which counts as zero). The sum of those fingers will result in a number that is either odd or even, so one of the players wins.
I tried to research whether the game is always mathematically fair and I’ve found conflicting explanations. My question is which of these, if any, is correct:
**Explanation number one** is that it depends on the rules. If you play with one hand per player, there are six possible numbers (0 through 5), so between the two players you have 11 possible sums (0 through 10) but 36 possible combinations of fingers held up, many of which arrive at the same sum. 18 combinations are odd and 18 combinations are even, so this version of the game would supposedly be fair.
However, some versions of the game consider the sum of 0 to be a draw, and in that case, there are more valid odd combinations (18) than even combinations (17, since a 0 hand with a 0 hand no longer counts) in the probability pool. And if the players choose to use *both* hands each, the possible sums are 0 through 20 and so there is a total of 121 combinations with 61 being even, unless the zero is considered a draw, in which case it’s 60-60.
**Explanation number two** is that the rules *don’t* matter and the game is fair no matter what. According to this explanation, the results are ultimately binary: it’s either odd or even, so with the two players you have four possible combinations: an even hand with an even hand (the result is an even sum), odd with odd (the result is even), odd with even (the result is odd) and even with odd (the result is odd). So two possible even combinations, two possible odd combinations, giving each player a 50% chance no matter the amount of hands or whether or not a zero sum is considered a draw.
My question is: which explanation is correct, and if neither is, what is the explanation? I have the mathematical prowess of a concussed goldfish so I need some help with this one.
In: 7
It’s a perfectly fair game. The numbers aren’t picked at random, their chosen by the players. So there is no assumption that the numbers will show up with the exact same frequency like the rolling dice. If the player decides to show five fingers as often as they do two or four that’s a strategic choice. But the results would be the same if the same if they never showed five and showed three twice as often. The only real choice is odd or even. And both players have the exact same choice and have an equal affect on the outcome.
For example, if I’m playing I only show one or two fingers because in my mind those represent the only two choices I have (odd or even).
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