Help with probability please

sackerpint1 · 2009-09-24 02:40:24

A scientist is about to compile a series of 11 similar programs. Suppose A_i is the event that the ith program compiles successfully for i = 1,...,11. When the programming task is simple, the scientist expects that 80% of programs will compile. When the programming task is hard, he expects that 40% of the programs will compile. Let B be the event that the programming task is simple and B complement be the event that the programming task is hard. The scientist believes that the events A_1,...A_11 are conditionally independent given B and given B complement. Let A be the event that exactly eight out of eleven programs compiled. What is the conditional probability of B given A?

Here's what I gather so far:

Conditional on B - The conditional probability that a particular collection of 8 programs out of 11 will compile is

(0.8)^8 * (0.2)^3 = 0.001342.

There are 11 choose 8 such collections of eight programs out of the 11. This is equal to 165, so the probability that exactly 8 programs will compile is 165 * 0.001342 = 0.2215.

Conditional on B complement - The conditional probability that a particular collection of 8 programs out of 11 will compile is

(0.4)^8 * (0.6)^3 = 0.0001416. This times 165 gives the probability of exactly 8 programs compiling. This value is equal to 0.02335.

I'm confused as to what to do to find the conditional probability of B given A, though. If anyone could help, it would be greatly appreciated!

mathsyperson · 2009-09-24 05:59:54

The question isn't quite answerable just yet.

You need to have the probabilities of two events:
(i) - 8 programs compiling successfully
(ii) - 8 programs compiling successfully, and the programs being hard.

Once you have them, you can divide (ii) by (i).
The probabilities you have calculated are correct, but to answer the question you also need the probability that the scientist will try to write a set of hard programs.

If this is q, then the final answer is given by

This probability will depend on q.
Trivially, q=0 --> P(B|A) = 0 and q=1 --> P(B|A) = 1.

If q=0.5, then P(B|A) ≈ 0.09165.

sackerpint1 · 2009-09-26 02:48:25

Thanks, mathsyperson! Apparently there was a problem with my textbook and it was supposed to include q, which is where I got confused.

Math Is Fun Forum

#1 2009-09-24 02:40:24

Help with probability please

#2 2009-09-24 05:59:54

Re: Help with probability please

#3 2009-09-26 02:48:25

Re: Help with probability please

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