A quality-control program at a plastic bottle production line involves inspecting finished bottles for flaws such as microscopic holes. The proportion of bottles that actually have such a flaw is only 0.0002, If a bottle has a flaw, the probability is 0.995 that is will fail the inspection. If a bottle does not have a flaw, the probability is 0.99 that it will pass the inspection.

a. If a bottle fails inspection ,what is the probability that it has a flaw?
b. If a bottle passes inspection, what is the probability that it does not have a flaw?

(a) If a bottle fails inspection, the probability that it has a flaw is P(F | Fail) = 0.0195

(b) If a bottle passes inspection, the probability that it does not have a flaw is P(NF | Pass) = 0.99

Step-by-step explanation:

Let F denote that the bottle has a flaw, NF denote that the bottle has no flaw, Pass denote that the bottle passes the inspection test and Fail denote that the bottle fails the inspection test.

We are given with the probabilities:

P(F) = 0.0002

P(Fail | F) = 0.995

P(Pass | NF) = 0.99

We can obtain the probabilities:

P(NF) = 1 - 0.0002 = 0.9998

P(Pass | F) = 1 - P(Fail | F)

                  = 1 - 0.995

P(Pass | F) = 0.005

P(Fail | NF) = 1 - P(Pass | NF)

                  = 1 - 0.99

P(Fail | NF) = 0.01

a. We need to find the probability that the bottle has a flaw given that it fails the inspection i.e. P(F | Fail). We can use the Baye's theorem to find this probability.

P(F | Fail) = P(Fail | F)*P(F) / [P(Fail | NF)*P(NF) + P(Fail | F)*P(F)]

               =  (0.995)*(0.0002)/[(0.01)*(0.9998) + (0.995)*(0.0002)]

               = 0.000199/(0.009998+0.000199)

               = 0.000199/0.010197

P(F | Fail) = 0.0195

b. We need to compute the probability that the bottle does not have a flaw given that it passes the inspection. i.e. P(NF | Pass)

P(NF | Pass) = P(Pass | NF)*P(NF) / [P(Pass | F)*P(F) + P(Pass | NF)*P(NF)]

                    = (0.99)*(0.9998)/[(0.005)*(0.0002) + (0.99)*(0.9998)]

                    = 0.989802/(0.000001 + 0.989802)

                    = 0.989802/0.989803

P(NF | Pass) = 0.99

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range:

2/3

step-by-step explanation: