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basketball probability question
you're a 40% shooter from 3 point range. over the span of 500 shots, what is the chance that you'll have a streak of 10 misses at some point?
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Re: basketball probability question (edited)
[ QUOTE ]
you're a 40% shooter from 3 point range. over the span of 500 shots, what is the chance that you'll have a streak of 10 misses at some point? [/ QUOTE ] The probability of missing 10 shots in a row at some time in N shots P(N) obeys a simple recurrence relation: P(N) = 0 for N < 10 P(10) = (.6)^10 P(11) = P(10) + (0.4)*(.6)^10 P(N) = P(N-1) + [1-P(N-11)]*(0.4)*(0.6)^10 for N >= 12. That is, the probability of it happening in the first N shots is the sum of the probability of it happening in the first N-1 plus the probability of it happening on the Nth shot after not happening earlier, but to happen on the Nth shot, it must NOT happen in the first N-11 shots, and then there must always be 1 basket followed by 10 misses. Soving this in Excel, P(500) = 70.55%. Edit: Originally did this with 60% shooter (and got 3.0%). |
#3
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Re: basketball probability question (edited)
[ QUOTE ]
[ QUOTE ] you're a 40% shooter from 3 point range. over the span of 500 shots, what is the chance that you'll have a streak of 10 misses at some point? [/ QUOTE ] The probability of missing 10 shots in a row at some time in N shots P(N) obeys a simple recurrence relation: P(N) = 0 for N < 10 P(10) = (.6)^10 P(11) = P(10) + (0.4)*(.6)^10 P(N) = P(N-1) + [1-P(N-11)]*(0.4)*(0.6)^10 for N >= 12. That is, the probability of it happening in the first N shots is the sum of the probability of it happening in the first N-1 plus the probability of it happening on the Nth shot after not happening earlier, but to happen on the Nth shot, it must NOT happen in the first N-11 shots, and then there must always be 1 basket followed by 10 misses. Soving this in Excel, P(500) = 70.55%. Edit: Originally did this with 60% shooter (and got 3.0%). [/ QUOTE ] Bruce, Is there a way to solve such recurrences in closed form? |
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Re: basketball probability question (edited)
[ QUOTE ]
[ QUOTE ] [ QUOTE ] you're a 40% shooter from 3 point range. over the span of 500 shots, what is the chance that you'll have a streak of 10 misses at some point? [/ QUOTE ] The probability of missing 10 shots in a row at some time in N shots P(N) obeys a simple recurrence relation: P(N) = 0 for N < 10 P(10) = (.6)^10 P(11) = P(10) + (0.4)*(.6)^10 P(N) = P(N-1) + [1-P(N-11)]*(0.4)*(0.6)^10 for N >= 12. That is, the probability of it happening in the first N shots is the sum of the probability of it happening in the first N-1 plus the probability of it happening on the Nth shot after not happening earlier, but to happen on the Nth shot, it must NOT happen in the first N-11 shots, and then there must always be 1 basket followed by 10 misses. Soving this in Excel, P(500) = 70.55%. Edit: Originally did this with 60% shooter (and got 3.0%). [/ QUOTE ] Bruce, Is there a way to solve such recurrences in closed form? [/ QUOTE ] I'm not Bruce, but you can set up a Markov Chain with 11 stages: Stage 1 = first shot or made the last shot Stage 2 = missed last shot, made the one before it Stage 3 = missed 2 in a row ... Stage 11 = missed 10 in a row for 1<=i<=10 P(going from stage i to stage i+1) = 0.6 (you miss 60% of the time, so you've now missed i in a row instead of i-1) P(going from stage i to stage 1) = 0.4 (you make the shot 40% of the time, so you've now missed 0 in a row) stage 11 is a sink, so P(going from 11 to 11) = 1 (once you've missed 10 in a row, you meet the criteria) So let A be a matrix with A(i,j) = P(going from stage i to stage j). Then A^n is a matrix with the property that A^n(i,j) is the probability of being in stage j n events after starting in stage i. Hence we want [A^500](1,11) = 0.7055 Dunno if that's what you were asking for or not [img]/images/graemlins/smile.gif[/img] Edit: And when I typed in the numbers backwards (make 60% shots), I got 0.0305 Also, here's some pseudo-code: A = 11 x 11 matrix for i=1,10 do A(i,1) = 0.6 A(i,i+1) = 0.4 end for B = A^500 print B(1,11) |
#5
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Re: basketball probability question (edited)
gimp,
that's exactly what i wanted. Markov chains are cool. |
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