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taylrl3
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Basically I have a Master’s degree in Astrophysics but I haven’t done any mathematics for a few years now and am getting a little rusty. I thought a good way of dusting off some of my probability skills would be to do an exercise. Here is the problem:
Suppose we want to model a game of pong based on 10 matches. The game is played between two players and is scored as first to three wins. An example dataset for this might be as follows:
ABBAA
BAAA
AABBA
BBB
BABB
AAA
AABA
BAAA
AABBB
AAA
I would like to estimate the probability of player A winning a match and am thinking about the best way in which to do so.
1. The first way could be to calculate the proportion of points won by player A in that match and then take the mean of those values across the ten matches as the estimate of the probability (calling the estimator p’). When this is calculated, p’ = 0.61.
2. Another way to define p’ would be as the total number of points won by player A in all of the 10 matches, divided by the total number of points played, however this gives an estimate of p as 0.6.
I am thinking about whether either of these is unbiased and whether there is any alternative which might be better for estimating the probability of A winning as I would prefer the estimator to be unbiased ideally. Also, would there be any benefit to a biased estimator?
I believe the second method of calculating the probability is the unbiased one and the first has a bias of 0.01 as the bias is defined as being the difference between an estimator and the expected value.
It would be great to hear anyones thoughts on this little problem and ways of doing things. I am also a little under confident when it comes to writing down the mathematical definitions of this problem at this point so any pointers on those lines would be great too.
Thanks in advance!
Taylrl
Suppose we want to model a game of pong based on 10 matches. The game is played between two players and is scored as first to three wins. An example dataset for this might be as follows:
ABBAA
BAAA
AABBA
BBB
BABB
AAA
AABA
BAAA
AABBB
AAA
I would like to estimate the probability of player A winning a match and am thinking about the best way in which to do so.
1. The first way could be to calculate the proportion of points won by player A in that match and then take the mean of those values across the ten matches as the estimate of the probability (calling the estimator p’). When this is calculated, p’ = 0.61.
2. Another way to define p’ would be as the total number of points won by player A in all of the 10 matches, divided by the total number of points played, however this gives an estimate of p as 0.6.
I am thinking about whether either of these is unbiased and whether there is any alternative which might be better for estimating the probability of A winning as I would prefer the estimator to be unbiased ideally. Also, would there be any benefit to a biased estimator?
I believe the second method of calculating the probability is the unbiased one and the first has a bias of 0.01 as the bias is defined as being the difference between an estimator and the expected value.
It would be great to hear anyones thoughts on this little problem and ways of doing things. I am also a little under confident when it comes to writing down the mathematical definitions of this problem at this point so any pointers on those lines would be great too.
Thanks in advance!
Taylrl
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