- #1
caffeine
I've been reviewing prob and stats which I took 10 years ago (ulp!) and came across a problem I can't solve. Consider the problem:
What is the expected minimum number of coin tosses you'd need to make in order to get 3 tails in a row?
Initial thought: the probability of getting a H followed by 3 T is p = (1/2)^4. These are Bernoulli trials, so the expected number of trials to get 3 T is:
E(T=3) = 1/p = 2^4 = 16
However, I've seen that the probability is really p = (1/2)^4 - 2, and the expected number of tosses to get 3 tails is really:
E(T=3) = 2^4 - 2 = 14
I wrote a program in C to calculate this expectation, and the average looks very 14-ish, definitely not 16-ish:
$ ./a.out
Average is 13.992740.
$ ./a.out
Average is 13.996880.
$ ./a.out
Average is 14.003285.
$ ./a.out
Average is 13.956330.
$ ./a.out
Average is 13.982135.
$ ./a.out
Average is 13.982135.
$ ./a.out
Average is 14.000490.
so it appears that 14 is correct. I'm really struggling to understand that "-2" in the probability and expectation value. Does anyone understand it?
What is the expected minimum number of coin tosses you'd need to make in order to get 3 tails in a row?
Initial thought: the probability of getting a H followed by 3 T is p = (1/2)^4. These are Bernoulli trials, so the expected number of trials to get 3 T is:
E(T=3) = 1/p = 2^4 = 16
However, I've seen that the probability is really p = (1/2)^4 - 2, and the expected number of tosses to get 3 tails is really:
E(T=3) = 2^4 - 2 = 14
I wrote a program in C to calculate this expectation, and the average looks very 14-ish, definitely not 16-ish:
$ ./a.out
Average is 13.992740.
$ ./a.out
Average is 13.996880.
$ ./a.out
Average is 14.003285.
$ ./a.out
Average is 13.956330.
$ ./a.out
Average is 13.982135.
$ ./a.out
Average is 13.982135.
$ ./a.out
Average is 14.000490.
so it appears that 14 is correct. I'm really struggling to understand that "-2" in the probability and expectation value. Does anyone understand it?
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