Question

A,B,C, and D represent events. Use the given probabilities to answer parts a - e.

P(A)=0.3
P(A∣C)=0.2
​

P(B)=0.25
P(B and C)=0.2
​

P(C)=0.4
​
a. Find P(B ′ ); that is, probability of B complement.
b. Assume events A and B are independent. Find P(A and B).
c. For the events A and C : i. Are the events A and C independent or dependent? Why? ii. Find P(A and C). Recall:
P(A)=0.3
P(A∣C)=0.2
​

P(B)=0.25
P(B and C)=0.2
​

P(C)=0.4
​
d. Find P(B∣C). e. Find P(B or C).

Answer 1

a. P(B') = 0.75
b. P(A and B) = 0.075
c. i. A and C are dependent. ii. P(A and C) = 0.08
d. P(B|C) = 0.5
e. P(B or C) = 0.45

Step-by-step explanation:

a. To find the probability of the complement of B, we subtract the probability of B from 1. P(B') = 1 - P(B) = 1 - 0.25 = 0.75.

b. If events A and B are independent, then P(A and B) = P(A) * P(B). Since P(A) = 0.3 and P(B) = 0.25, P(A and B) = 0.3 * 0.25 = 0.075.

c. i. Events A and C are dependent because the conditional probability P(A|C) is not equal to the marginal probability P(A).

ii. To find P(A and C), we use the formula P(A and C) = P(A|C) * P(C). Since P(A|C) = 0.2 and P(C) = 0.4, P(A and C) = 0.2 * 0.4 = 0.08.

d. To find the probability of B given C, we use the formula P(B|C) = P(B and C) / P(C). Since P(B and C) = 0.2 and P(C) = 0.4, P(B|C) = 0.2 / 0.4 = 0.5.

e. To find the probability of B or C, we use the formula P(B or C) = P(B) + P(C) - P(B and C). Since P(B) = 0.25, P(C) = 0.4, and P(B and C) = 0.2, P(B or C) = 0.25 + 0.4 - 0.2 = 0.45.