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#1 2005-09-14 08:50:39
- cubix
- Guest
sequential experiments: probability
This probelm concerns random number generation. The gist of the probelm is that 10 numbers are selected from the interval [0,1], and goal is to discover the probabilty that the first five numbers are less than 1/4 and the last five numbers are greater than 1/2.
I'm stuck on what to do this kind of problem. I know how to find the probability of a number greater thanx or less than x, but I'm lost with how to work with a problem that involves the sequence.
#2 2005-09-14 09:38:00
- kylekatarn
- Power Member
Offline
Re: sequential experiments: probability
your random number generator bounds are ø=[0,1]
let the generator return a number, r : rєø
now consider:
A:"what's the probability of 0<r<1"
I answer: " P(A)=1"
B:"what's the probability of 0<r<=1/2
I answer:"P(B)=1/2"
We can see this as a "geometrical probability" problem. A bigger domain implies greater probability.
so lets consider the first number generated: g1
P(g1;"less than 1/4")=1/4
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P(gN," less than k")=k ; 0<k<1
But P(gN,"greater than b")=1-P(gN,"less than b)
Its a random number gen. so each gN is independent. You want first 5 to be < 1/4
P(g1;"less than 1/4") * ... * P(g5;"less than 1/4") = (1/4)^5 = 1/1024
The last 5 must be >1/2
P(g6,"greater than 1/2) * ... * P(g10,"greater than 1/2) =
= [1-P(g6,"less than 1/2)] * ... * [1-P(g10,"less than 1/2)] =
= [1-1/2]^5 = (1/2)^5 =1/32
We multiply the two results (P1 and P2)
1/1024 * 1/32 = 1/32768
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This is just an idea.
Last edited by kylekatarn (2005-09-14 09:40:17)