Question

question 4
eight people were asked if they have a brother or sister. this venn diagram shows the results. a person is randomly chosen from those shown in the venn diagram.
let event a = the person has a sister.
let event b = the person has a brother.

what does $p(b|a)=0.25$ mean in terms of the venn diagram?
a. of the 4 people who have a sister, 1 of them also has a brother.
b. of the 2 people that have brothers, 1 of them also has a sister.
c. of the 8 people surveyed, 2 of these have a brother.
d. of the 4 people who dont have a sister, 1 of them has a brother.

question 5
$p(a)=0.50$
$p(b)=0.80$
$p(a \text{ and } b)=0.20$
what is $p(b|a)$?
a. 0.40
b. 0.30
c. 0.25
d. 0.80

question 6
at a high school, the probability that a student is a senior is 0.25. the probability that a student plays a sport is 0.15. the probability that a student is a senior and plays a sport is 0.10.
what is the probability that a randomly selected student plays a sport, given that the student is a senior?
a. 0.40
b. 0.66
c. 0.15
d. 0.10

Explanation:

Question 4

$P(B|A)$ is the conditional probability of event $B$ (has a brother) given event $A$ (has a sister). First, count people with a sister: 4 (Omar, Davis, Karl, Lyle). Of these, only 1 (Omar) also has a brother, so $\frac{1}{4}=0.25$, which matches the value.

Step1: Recall conditional probability formula

The formula for conditional probability is $P(B|A)=\frac{P(A \text{ and } B)}{P(A)}$

Step2: Substitute given values

Substitute $P(A \text{ and } B)=0.20$ and $P(A)=0.50$:
$P(B|A)=\frac{0.20}{0.50}$

Step3: Calculate the result

$\frac{0.20}{0.50}=0.40$

Step1: Define events and formula

Let $A$ = student is a senior, $B$ = student plays a sport. Use $P(B|A)=\frac{P(A \text{ and } B)}{P(A)}$

Step2: Substitute given values

Substitute $P(A \text{ and } B)=0.10$ and $P(A)=0.25$:
$P(B|A)=\frac{0.10}{0.25}$

Step3: Calculate the result

$\frac{0.10}{0.25}=0.40$

Answer:

A. Of the 4 people who have a sister, 1 of them also has a brother.

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Question 5