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#1 2005-09-13 10:50:39

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 2005-09-13 11:38:00

kylekatarn
Member
Registered: 2005-07-24
Posts: 445

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
.
.
.
.
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

------------------------------------------------------
This is just an idea.

Last edited by kylekatarn (2005-09-13 11:40:17)

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