Question

the table shows the number of male and female students enrolled in nursing at the university of oklahoma health sciences center for a recent semester.

	nursing majors	non - nursing majors	total
females	725	1682	2407
total	819	2786	3605

(a) find the probability that a randomly selected student is male, given that the student is a nursing major.
(b) find the probability that a randomly selected student is a nursing major, given that the student is male.

1. in a sample of 1000 u.s. adults, 180 dine out at a restaurant more than once per week. two u.s. adults are selected at random without replacement.

(a) find the probability that both adults dine out more than once per week.
(b) find the probability that neither adult dines out more than once per week.
(c) find the probability that at least one of the two adults dines out more than once per week.
(d) which of the events can be considered unusual? explain

1. of the cans produced by a company, 96% do not have a puncture, 93% do not have a smashed edge, and 89.3% do not have a puncture and do not have a smashed edge. find the probability that a randomly selected can does not have a puncture or does not have a smashed edge.

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In the context of frequency - tables, if we want to find the probability that a student is male given that the student is a nursing major, we use the formula $P(\text{Male}|\text{Nursing})=\frac{\text{Number of male nursing majors}}{\text{Total number of nursing majors}}$.

Step2: Calculate the probability for 8(a)

We have 94 male nursing majors and 819 total nursing majors. So $P(\text{Male}|\text{Nursing})=\frac{94}{819}\approx0.115$.

Step3: Calculate the probability for 8(b)

To find the probability that a student is a nursing major given that the student is male, we use the formula $P(\text{Nursing}|\text{Male})=\frac{\text{Number of male nursing majors}}{\text{Total number of males}}$. There are 94 male nursing majors and 1198 total males. So $P(\text{Nursing}|\text{Male})=\frac{94}{1198}\approx0.078$.

Step4: Calculate the probability for 9(a)

The probability that the first adult dines out more than once per week is $\frac{180}{1000}$. Since the selection is without replacement, the probability that the second adult dines out more than once per week given that the first one does is $\frac{179}{999}$. The probability that both dine out more than once per week is $\frac{180}{1000}\times\frac{179}{999}=\frac{179}{5550}\approx0.0323$.

Step5: Calculate the probability for 9(b)

The number of adults who do not dine out more than once per week is $1000 - 180=820$. The probability that the first adult does not dine out more than once per week is $\frac{820}{1000}$, and the probability that the second one does not given that the first one does not is $\frac{819}{999}$. So the probability that neither dines out more than once per week is $\frac{820}{1000}\times\frac{819}{999}=\frac{3731}{5550}\approx0.6723$.

Step6: Calculate the probability for 9(c)

The probability that at least one of the two adults dines out more than once per week is $1 - P(\text{neither dines out more than once per week})$. So it is $1-\frac{3731}{5550}=\frac{1819}{5550}\approx0.3277$. Another way is $P(\text{first dines and second doesn't})+P(\text{first doesn't and second dines})+P(\text{both dine})=\frac{180}{1000}\times\frac{820}{999}+\frac{820}{1000}\times\frac{180}{999}+\frac{180}{1000}\times\frac{179}{999}\approx0.3277$.

Step7: Calculate the probability for 10

We use the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Let $A$ be the event that a can does not have a puncture and $B$ be the event that a can does not have a smashed edge. We know $P(A) = 0.96$, $P(B)=0.93$ and $P(A\cap B)=0.893$. Then $P(A\cup B)=0.96 + 0.93-0.893=0.997$.

Answer:

8(a): $\frac{94}{819}\approx0.115$
8(b): $\frac{94}{1198}\approx0.078$
9(a): $\frac{179}{5550}\approx0.0323$
9(b): $\frac{3731}{5550}\approx0.6723$
9(c): $\frac{1819}{5550}\approx0.3277$
10: $0.997$