A new computer program consists of two modules The first mod

A new computer program consists of two modules. The first module contains an error with probability 0.2. The second module is more complex, it has a probability of 0.4 to contain an error, independently of the first module. An error in the first module alone causes the program to crash with probability 0.5. For the second module alone, this probability is 0.8. If there are errors in both modules, the program crashes with probability 0.9.

a)What is the probability of a crash?

b)Suppose the program crashed. What is the probability of errors in both modules?

Can anyone with at least 1000 answers solve this question step by step..?

Thanx!

Solution

Since both the modules are independent of each other,

P(Error in both the modules) = P(Error in first module)*P(Error in 2nd module) = 0.2*0.4 = 0.08

Given,

P(Error in 1st) = 0.2

P(Error in 2nd) = 0.4

P(Crash| Error in 1st) = 0.5

P(Crash| Error in 2nd) = 0.8

P(Crash| Both errors) = 0.9

a) P(Crash) = P(Crash| Error in 1st)*P(Error in 1st) + P(Crash| Error in 2nd)*P(Error in 2nd) + P(Crash|Both errors)*P(Both errors)

= 0.2*0.5 + 0.4*0.8 + 0.08*0.9

= 0.492

b) P(Both errors| Crash)

= (P(Crash|Both errors)*P(Both errors)) / P(Crash)

= (0.08*0.9)/0.492

= 0.1463 [Rounded off to 4 decimal places]