Answer:
a) P ( A & B ) = 0.1995
b) P (A U B ) = 0.7905
c) P (A/B) = 0.2625
d) P(B/A') = 0.194805
e) NOT mutually exclusive
f) NOT Independent
Step-by-step explanation:
Declare Events:
- buying gasoline = Event A
- buying groceries = Event B
Given:
- P(A) = 0.23
- P(B) = 0.76
- P(B/A) = 0.85
Find:
- a. Find the probability that a typical customer buys both gasoline and groceries.
- b. Find the probability that a typical customer buys gasoline or groceries.
- c. Find the conditional probability of buying gasoline given that the customer buys groceries.
- d. Find the conditional probability of buying groceries given that the customer did not buy gasoline.
- e Are these two events (groceries, gasoline) mutually exclusive?
- f Are these two events independent?
Solution:
- a) P ( A & B ) ?
P ( A & B ) = P(B/A) * P(A) = 0.85*0.23 = 0.1995
- b) P (A U B ) ?
P (A U B ) = P(A) + P(B) - P(A&B)
P (A U B ) = 0.23 + 0.76 - 0.1995
P (A U B ) = 0.7905
- c) P ( A / B )?
P ( A / B ) = P(A&B) / P(B)
= 0.1995 / 0.76
= 0.2625
- d) P( B / A') ?
P( B / A') = P ( B & A') / P(A')
P ( B & A' ) = 1 - P( A / B) = 1 - 0.85 = 0.15
P ( B / A' ) = 0.15 / (1 - 0.23)
= 0.194805
- e) Are the mutually exclusive ?
The condition for mutually exclusive events is as follows:
P ( A & B ) = 0 for mutually exclusive events.
In our case P ( A & B ) = 0.1995 is not zero.
Hence, NOT MUTUALLY EXCLUSIVE
- f) Are the two events independent?
The condition for independent events is as follows:
P ( A & B ) = P (A) * P(B) for mutually exclusive events.
In our case,
0.1995 = 0.23*0.76
0.1995 = 0.1748 (NOT EQUAL)
Hence, NOT INDEPENDENT
