Question 363026
let x = lens defect
let y = crack defect


p(x) = .12


p(y) = .29


p(x+y) = .07


the probability of p(x) without p(x+y) should be equal to p(x) - p(x + y) = .12 - .07 = .05


the probability of p(x) without p(x+y) should be equal to p(y) - p(x + y) = .29


we'll examine these with a hypothetical situation to see what's happening.


the probability of x occurring is .12.


this means that, if the probabilities hold, and you sell 1000 products, then:


12% of them will have a lens defect.
29% of them will have a crack defect.


12% of 1000 = 120
29% of 1000 = 290


total defective products are therefore 390.


of these .07 * 1000 = 70 have both a crack defect and a lens defect.


to find how many products have only a lens defect or only a crack defect, you have to subtract 70 from both to get:


50 have only a lens defect.
220 have only a crack defect.


50 / 1000 = .05
220 / 1000 = .22


.05 = .12 - .05 which is the same as p(x) - p(x + y)
.22 = .29 - .05 which is the same as p(y) - p(x + y)


the formulas look good.


the question is, of the total products sold, how many have only a lens defect or only a crack defect.


50 have only a lens defect and 220 have only a crack defect which makes a total of 270 with only a lens defect or only a crack defect.


270 / 1000 = .27 which means that 27% of the time this occurs which means that the probability of only a lens defect or only a crack defect = .27.


that should be your answer.


to go one step further, let's take the probability of a product having a defect.


this could be either lens only or crack only or both.


the equation for that is:


p(x or y) = p(x) + p(y) - p(x + y)


that comes out to be .12 + .29 - .07 = .34


34% of the time a product will have a defect.


it will be either a lens only or a crack only or a lens and a crack.


you have to subtract p(x + y) because (x + y) is both a memmber of x and a member of y.


if you want to know the probability that it will have a lens only or a crack only, then you have to subtract products that have a lens defect and a crack defect.


the formula becomes p(x or y) - p(x + y) = p(x) + p(y) - p(x + y) - p(x + y) which becomes:


p(x or y) - p(x + y) = p(x) + p(y) - 2*p(x + y)


so we have:


p(x or y) - p(x + y) = .12 + .29 - 2*.07 = .41 - .14 = .27


the hypothetical numbers confirm the probabilities as being correct.