Question

1. (6 points) the probability that a student studies is .7. given that she studies, the probability is .8 that she will pass the course. given that she does not study, the probability is .3 that she will pass the course. find the probability that (a) she will study and pass the course. (b) she will not study and will pass the course. (c) she will pass the course.

Explanation:

Step1: Define probabilities

Let $P(S) = 0.7$ be the probability of studying, $P(\overline{S})=1 - 0.7 = 0.3$ be the probability of not - studying, $P(P|S)=0.8$ be the probability of passing given studying, and $P(P|\overline{S}) = 0.3$ be the probability of passing given not - studying.

Step2: Calculate probability of studying and passing (a)

Use the formula for conditional probability $P(A\cap B)=P(A|B)P(B)$. Here, for studying and passing, $P(S\cap P)=P(P|S)P(S)$.
$P(S\cap P)=0.8\times0.7 = 0.56$

Step3: Calculate probability of not - studying and passing (b)

Use the formula $P(\overline{S}\cap P)=P(P|\overline{S})P(\overline{S})$.
$P(\overline{S}\cap P)=0.3\times0.3=0.09$

Step4: Calculate probability of passing (c)

Use the law of total probability $P(P)=P(P|S)P(S)+P(P|\overline{S})P(\overline{S})$.
$P(P)=0.8\times0.7 + 0.3\times0.3=0.56+0.09 = 0.65$

Answer:

(a) $0.56$
(b) $0.09$
(c) $0.65$