Om is a lead engineer in charge of a team of 8 electrical engineers. His job is to design a cellu- lar phone (handset). The handset consists of 4 components - antenna, circuitry, LCD screen, and software (a) In how many different ways can Tom form 4 groups of 2 engineers in each for designing the 4 components?(b) The overall design of the handset put together by Tom's team will be flawed with probability 0.2. If the design is flawed, the phone will not be working after a week with probability 0.8. If the design is not flawed, it will continue to work after a week with probability 0.9. Compute the probability that the phone will be working after a week.(c) (Continuing from part (b)) Given that the phone was still working after a week, what is the probability that the design was flawed?

a) there are 2520 possible groups of engineers

b) the probability is 0.76 (76%)

c) the probability is 1/19 ( 5.26%)

Step-by-step explanation:

a) the possible combinations when there is grouping of elements is

C= n!/(a!*b!. ...) , where n= total size and a,b,... = are the group sizes of a ,b and others

in our case , for 8 elements in 4 groups of 2 is

C = 8!/(2!*2!*2!*2!) = 8!/2⁴ = 2520 possible groups of engineers

b) defining the event W= the cellphone works after a week , then the probability is

P(W) = probability that the cellphone is flawed *  probability that the cellphone works after a week given that the cellphone is flawed + probability that the cellphone is not flawed *  probability that the cellphone works after a week given that the cellphone is not flawed = 0.2 * 0.2 + 0.8 * 0.9 = 0.76

c) the conditional probability can be calculated with the theorem of Bayes. Defining the event F= the design is flawed. Then we have

P(F/W)= P(F∩W)/P(W) = 0.2 * 0.2/0.76 = 1/19 ( 5.26%)

where

P(F∩W) = probability that the design is flawed and the cellphone works after a week

P(F/W)= probability that the design is flawed given the cellphone was still working after a week