12.34 Screening job applicants. A company retains a psychologist to assess whether job applicants are suited for assembly-line work. The psychologist classifies applicants as one of A (well suited), B (marginal), or C (not suited). The company is concerned about the event D that an employee leaves the company within a year of being hired. Data on all people hired in the past five years give these probabilities:
P(A) = 0.4 P(B) = 0.3 P(C) = 0.3
P(A and D) = 0.1 P(B and D) = 0.1 P(C and D) = 0.2
What is P(D), the probability that an employee leaves within a year?
I am not sure if I just add up all of the intersections to get .4, or do I take (.4*.1) + (.3*.1) + (.3*.2)=.13?
.1 + .1 + .2 = .4
Thank you!
Welcome!
Well, probability can be a bit tricky sometimes, just like trying to figure out if a clown's shoe is bigger than the other. But don't worry, I've got you covered!
To find the probability that an employee leaves within a year (P(D)), we can use the Law of Total Probability. This law states that the probability of an event (D) can be calculated by summing up the probabilities of that event happening given each possible condition.
In this case, we have three possible conditions: A, B, and C. So, we need to calculate the probability of D given each of these conditions and then add them up.
Let's do the calculations:
P(D|A) = 0.1 (probability of leaving within a year given the employee is well suited)
P(D|B) = 0.1 (probability of leaving within a year given the employee is marginal)
P(D|C) = 0.2 (probability of leaving within a year given the employee is not suited)
Now, we multiply each of these conditional probabilities by the probability of each condition and add them up:
P(D) = P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C)
= (0.1 * 0.4) + (0.1 * 0.3) + (0.2 * 0.3)
= 0.04 + 0.03 + 0.06
= 0.13
So, P(D) is indeed 0.13. Just remember, when dealing with probability, it's important to juggle the numbers carefully.
To find the probability that an employee leaves within a year (P(D)), you need to consider all the possible ways in which this event can occur. In this case, an employee can leave within a year if they are classified as A and leave (A and D), if they are classified as B and leave (B and D), or if they are classified as C and leave (C and D).
So, to calculate P(D), you need to add up the probabilities of each of these individual events occurring. You can use the formula:
P(D) = P(A and D) + P(B and D) + P(C and D)
Now let's substitute the given probabilities into the formula:
P(D) = 0.1 + 0.1 + 0.2
P(D) = 0.2 + 0.1
P(D) = 0.3
Therefore, the probability that an employee leaves within a year (P(D)) is 0.3 or 30%.
So, in this case, you should add up the probabilities of each of the intersections to get the correct answer.