A manager needs to select a group of 4 research associates (RAs) to form a research team. Out of a pool of 16 total RAs, 10 are female and 6 are male. If the manager randomly selects 4 RAs from this pool, what is the probability that the research team will include 2 females and 2 males?
The total number of outcomes in the sample space,
N = C
16 
4
16!

1,820 .
The number of ways to select 2 females from the 10 available, C
10 
2
10!

45 .
The number of ways to select 2 males from the 6 available, C
6 
2
6!

15 .
The number of outcomes that satisfy the condition of 2 females and 2 males is,
N
A
= 45 × 15 = 675.
Therefore, the probability that a randomly selected set of 4 will include 2 females and 2 males is
P
A

A
10 6
N C C
2
N

C
16
4
2

1,820

.3709
.
In a recent survey about the direction of the U.S. economy, a sample was asked whether they thought the U.S. economy was headed in the “right direction” or the “wrong direction” (the only two choices) Of those surveyed, 41% of the respondents said that they thought the U.S. economy was headed in the right direction. Males comprised 56% of the sample, and of the males, 35% said the U.S. economy was headed in the right direction. A person is randomly chosen from the survey sample.
Let M denote Male, F denote Female, R denote “right direction” and W denote “wrong direction.” Given: P(M) = 0.56, P(R) = 0.4, P(R|M) = 0.35
M
F
Total
R W Total
P(R∩M) P(W∩M)
0.56
P(R∩F) P(W∩F) P(F)
0.41 P(W) 1
What is the probability that the person we select is male that thinks the U.S. economy is headed in the right direction?
Looking for P(R ∩ M) = P(R|M)P(M) = (0.35)(0.56) = 0.196

A publisher sends advertising materials for an accounting text to 80% of all professors teaching the appropriate course. Thirty percent (30%) of the professors who received this material adopted the book, as did 10% of the professors who did not receive the material. What is the probability that a professor who adopts the book has received the advertising material?
A production manager knows that 5% of components produced by a particular manufacturing process have some defect. A random sample of 6 components is selected.
(a) What is the probability that none of the 6 components have a defect?
(b) What is the probability that one of the components has a defect?
(c) What is the probability that at least 2 of the components have a defect?
A customer service center receives, on average, 4.2 calls per minute. If the distribution of calls is Poisson, what is the probability of receiving at least 3 calls during a particular minute?
What is the probability of receiving between 3 and 5 calls (inclusive) in a given minute?
Grocery Store buys chickens from Aggie Farms and t-sip Foods. Among the chickens from
Aggie Farms, 10% of the processed chickens weigh less than 3 pounds. Among the chickens from t-sip Foods, 20% weight less than 3 pounds. Grocery Store buys 40% of its chicken from
Aggie Farms and mixes the products together with no identification of the supplier.
Suppose you purchase a chicken from Grocery Store that weighs more than 3 pounds.
What is the probability that it came from Aggie Farms?