"Once or two out of three?" Topic

Hi All,

Probabilities not being my forte, can anyone tell me if it is easier to win once or two out of three at say a dice-off, coin flip, guessing the closest number between one and 10, &c.?

I've thought about it for a while and looked online as well to no avail.

Thanks for any insights.

Yours,

Legion

If both players have an equal chance of winning a single around (50-50), then the chances of winning exactly 2 rounds out of 3 are:

0.5*0.5*0.5*3 = 0.375

The chances of winning exactly 3 rounds out of 3 = 0.5*0.5*0.5 = 0.125

The chance of winning exactly one round = 0.375 (symmetry with 1st case)

The chance of winning 0 rounds = 0.125 (symmetry with 2nd case)

But maybe this isn't what you're asking for?

2 out of 3 and a single toss ends up about the same.

a single toss, each side has a 50/50 chance of winning.

when there are 2 people, with 3 tosses, each one has a 50 percent chance of winning. (this continues no matter how many times, as long as you use an odd number 3 out of 5, 4 out of 7, etc…) There are more variations with more flips, sometimes winning 5 of 5 instead of just 3 of 5, but the chance of winning at least 3 is still roughly 50%.

for the math.
With 3 rolls there are four possible outcomes. with the following ways of them happening

0 of the tosses 1 way this can happen (lose lose lose)
1 of the tosses 3 ways (win lose lose, lose win lose, lose lose win)
2 of the tosses 3 ways (win win lose, win lose win, lose win win)
3 of the tosses 1 (win win win)

4 ways you can win overall, 4 ways you can lose overall. So a 50% chance.

Even numbers are slightly different, winning 2 out of 4, there are some cases where each side could win exactly twice – if that results in a roll off then the odds go back to 50%, if not then the tie has to be accounted for so each persons odds gets closer and closer to 50% (without ever actually making it to 50%) the more times you roll.

I think you need to be a bit more specific with the question.

But on the coin toss, if you toss a coin three times the possible outcomes are :
WWW
WWL
WLW
WLL
LWW
LWL
LLW
LLL

Each of these outcome has an equal (0.125) probablity. So the probabilty of winning exactly two out of three is 3 x 0.125 = 0.375.
The probability of winning two or more times is 0.5.

Thanks guys.

Very helpful.

Legion