Consider the following probabilities: P(Ac) = 0.57, P(B) = 0.36, and P(A ∩ Bc) = 0.03.
a. Find P(A | Bc). (Do not round intermediate calculations. Round your answer to 2 decimal places.)
b. Find P(Bc | A). (Do not round intermediate calculations. Round your answer to 3 decimal places.)
c. Are A and B independent events?
multiple choice
Yes because P(A | Bc) = P(A).
Yes because P(A ∩ Bc) ≠ 0.
No because P(A | Bc) ≠ P(A).
No because P(A ∩ Bc) ≠ 0.
a. P(A | Bc) can be calculated using the formula:
P(A | Bc) = P(A ∩ Bc) / P(Bc)
Given that P(A ∩ Bc) = 0.03 and P(Bc) = 1 - P(B) = 1 - 0.36 = 0.64, we can substitute these values into the formula:
P(A | Bc) = 0.03 / 0.64 ≈ 0.0469
Therefore, P(A | Bc) ≈ 0.05 (rounded to 2 decimal places).
b. P(Bc | A) can be calculated using the formula:
P(Bc | A) = P(A ∩ Bc) / P(A)
Given that P(A ∩ Bc) = 0.03 and P(A) = 1 - P(Ac) = 1 - 0.57 = 0.43, we can substitute these values into the formula:
P(Bc | A) = 0.03 / 0.43 ≈ 0.0698
Therefore, P(Bc | A) ≈ 0.070 (rounded to 3 decimal places).
c. A and B are independent events if and only if P(A | Bc) = P(A).
From the calculations above:
P(A | Bc) ≈ 0.05
P(A) = 0.43
Since P(A | Bc) ≠ P(A), we can conclude that A and B are not independent events.
The answer is: No because P(A | Bc) ≠ P(A).
To find the probabilities P(A | Bc) and P(Bc | A), we can use the formula for conditional probability:
P(A | B) = P(A ∩ Bc) / P(Bc)
P(Bc | A) = P(A ∩ Bc) / P(A)
a. To find P(A | Bc), we can use the given values:
P(Ac) = 0.57
P(B) = 0.36
P(A ∩ Bc) = 0.03
We know that P(A) is the complement of P(Ac), which means P(A) + P(Ac) = 1.
Using this information, we can calculate P(A):
P(A) = 1 - P(Ac) = 1 - 0.57 = 0.43
Now we can substitute the values into the formula:
P(A | Bc) = P(A ∩ Bc) / P(Bc) = 0.03 / (1 - P(B)) = 0.03 / (1 - 0.36) = 0.03 / 0.64 ≈ 0.0469
Therefore, P(A | Bc) is approximately 0.0469.
b. To find P(Bc | A), we can use the given values:
P(Ac) = 0.57
P(B) = 0.36
P(A ∩ Bc) = 0.03
Using the same logic as before, we can find P(B):
P(B) = 1 - P(Bc) = 1 - P(A ∩ Bc) = 1 - 0.03 = 0.97
Now we can substitute the values into the formula:
P(Bc | A) = P(A ∩ Bc) / P(A) = 0.03 / P(A) = 0.03 / 0.43 ≈ 0.0698
Therefore, P(Bc | A) is approximately 0.0698.
c. To determine if A and B are independent events, we need to check if P(A | Bc) equals P(A). Comparing the values from part a, we can see that P(A | Bc) ≈ 0.0469 and P(A) = 0.43.
Since P(A | Bc) ≠ P(A), A and B are not independent events.
Therefore, the correct answer is: No because P(A | Bc) ≠ P(A).
To find the answers to these questions, we need to understand some concepts in probability and use the given probabilities.
a. To find P(A | Bc), we can use the formula for conditional probability:
P(A | Bc) = P(A ∩ Bc) / P(Bc)
We are given P(A ∩ Bc) = 0.03 and need to find P(Bc).
To find P(Bc), we can use the complement rule:
P(Bc) = 1 - P(B)
Given P(B) = 0.36, we can find P(Bc) = 1 - 0.36 = 0.64.
Now we can calculate P(A | Bc):
P(A | Bc) = P(A ∩ Bc) / P(Bc)
P(A | Bc) = 0.03 / 0.64
Calculating this division, P(A | Bc) ≈ 0.046875. Rounding to two decimal places, the answer is 0.05.
Therefore, P(A | Bc) ≈ 0.05.
b. To find P(Bc | A), we can use the formula for conditional probability:
P(Bc | A) = P(A ∩ Bc) / P(A)
We are given P(A ∩ Bc) = 0.03 and need to find P(A).
To find P(Ac), we can use the complement rule:
P(Ac) = 1 - P(A)
Given P(Ac) = 0.57, we can find P(A) = 1 - 0.57 = 0.43.
Now we can calculate P(Bc | A):
P(Bc | A) = P(A ∩ Bc) / P(A)
P(Bc | A) = 0.03 / 0.43
Calculating this division, P(Bc | A) ≈ 0.069767. Rounding to three decimal places, the answer is 0.070.
Therefore, P(Bc | A) ≈ 0.070.
c. To determine if events A and B are independent, we need to check if the conditional probability P(A | B) is equal to the marginal probability P(A).
If P(A | B) = P(A), then events A and B are independent.
However, the question asks if A and B are independent based on the options provided. Let's analyze the options:
- Yes because P(A | Bc) = P(A): This option does not provide any information about the probability of A and B specifically.
- Yes because P(A ∩ Bc) ≠ 0: This option does not provide any information about the probability of A and B specifically.
- No because P(A | Bc) ≠ P(A): This option suggests that A and B are not independent because the conditional probability P(A | Bc) is not equal to the marginal probability P(A). This could be the correct answer.
- No because P(A ∩ Bc) ≠ 0: This option suggests that A and B are not independent because the intersection probability P(A ∩ Bc) is not equal to zero. This could also be the correct answer.
Based on these options, either option "No because P(A | Bc) ≠ P(A)" or option "No because P(A ∩ Bc) ≠ 0" could be the correct answer to determine that A and B are not independent. The correct option may depend on the context or information not provided in the question.