Question

Consider a sample space S, and two events A and B such that P(A ∩ B) = 0.2, P(A ∪ B) = 0.6, P(B ∪ ̄A) = 0.8 (a) [0.5 points] Calculate P (A). (b) [0.5 points] Calculation P (B)

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Clarabelle

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To solve this problem, we can use the concept of set theory and probability.

Step 1: Let's break down the given information using the properties of set theory and probability.

Given:

P(A ∩ B) = 0.2

P(A ∪ B) = 0.6

P(B ∪ ̄A) = 0.8

Step 2: We can use the formula for the probability of the union of two events to find P(A ∪ B).

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Given that P(A ∪ B) = 0.6 and P(A ∩ B) = 0.2, we can substitute these values in the equation and solve for P(A ∪ B):

0.6 = P(A) + P(B) - 0.2

Step 3: Rearrange the equation to solve for P(A):

P(A) = 0.6 - P(B) + 0.2

Step 4: Now, we need to find P(B ∪ ̄A). The complement of event A is denoted by ̄A, which represents all the outcomes in the sample space S that are not in A.

Using the formula for the probability of the union of two events, we have:

P(B ∪ ̄A) = P(B) + P(̄A) - P(B ∩ ̄A)

We can simplify this equation to:

P(B ∪ ̄A) = P(B) + P(̄A) - P(B ∩ ̄A) = 0.8

Step 5: Since we want to find P(A) and P(B), we need to eliminate P(̄A) and P(B ∩ ̄A). To do this, we can use the complement rule and rewrite P(B ∪ ̄A) in terms of P(A ∪ B) as follows:

P(B ∪ ̄A) = 1 - P(A ∪ B)

Substituting the given value of P(A ∪ B) = 0.6, we have:

0.8 = 1 - 0.6

Step 6: Now we can solve for P(B):

0.8 = 1 - P(A ∪ B) + P(B) = 0.4 + P(B)

P(B) = 0.8 - 0.4 = 0.4

Step 7: Finally, substitute the value of P(B) into the equation for P(A) that we obtained earlier:

P(A) = 0.6 - P(B) + 0.2 = 0.6 - 0.4 + 0.2 = 0.4

Answer:

(a) P(A) = 0.4

(b) P(B) = 0.4

Step 1: Let's break down the given information using the properties of set theory and probability.

Given:

P(A ∩ B) = 0.2

P(A ∪ B) = 0.6

P(B ∪ ̄A) = 0.8

Step 2: We can use the formula for the probability of the union of two events to find P(A ∪ B).

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Given that P(A ∪ B) = 0.6 and P(A ∩ B) = 0.2, we can substitute these values in the equation and solve for P(A ∪ B):

0.6 = P(A) + P(B) - 0.2

Step 3: Rearrange the equation to solve for P(A):

P(A) = 0.6 - P(B) + 0.2

Step 4: Now, we need to find P(B ∪ ̄A). The complement of event A is denoted by ̄A, which represents all the outcomes in the sample space S that are not in A.

Using the formula for the probability of the union of two events, we have:

P(B ∪ ̄A) = P(B) + P(̄A) - P(B ∩ ̄A)

We can simplify this equation to:

P(B ∪ ̄A) = P(B) + P(̄A) - P(B ∩ ̄A) = 0.8

Step 5: Since we want to find P(A) and P(B), we need to eliminate P(̄A) and P(B ∩ ̄A). To do this, we can use the complement rule and rewrite P(B ∪ ̄A) in terms of P(A ∪ B) as follows:

P(B ∪ ̄A) = 1 - P(A ∪ B)

Substituting the given value of P(A ∪ B) = 0.6, we have:

0.8 = 1 - 0.6

Step 6: Now we can solve for P(B):

0.8 = 1 - P(A ∪ B) + P(B) = 0.4 + P(B)

P(B) = 0.8 - 0.4 = 0.4

Step 7: Finally, substitute the value of P(B) into the equation for P(A) that we obtained earlier:

P(A) = 0.6 - P(B) + 0.2 = 0.6 - 0.4 + 0.2 = 0.4

Answer:

(a) P(A) = 0.4

(b) P(B) = 0.4

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