Suppose P( A given B)= .6, P(at least one of the two events occurs)= .8, and P(exactly one of the two events occurs)=.6. Then P(A) is equal to?
The answer is 2/3. Could someone please explain how to get it. Thanks so much!
To find the value of P(A), we can use the concept of conditional probability and the given information.
First, note that "at least one of the two events occurs" can be interpreted as the union of Event A and Event B (A ∪ B). Therefore, we have:
P(A ∪ B) = 0.8
Next, we can express "exactly one of the two events occurs" as the exclusive OR (XOR) of Event A and Event B (A ⊕ B). This means that either A occurs or B occurs, but not both. Therefore, we can write:
P(A ⊕ B) = 0.6
Now, we can use the concept of XOR to break down P(A ∪ B) into the sum of P(A) and P(B) minus twice the probability of both A and B occurring (P(A ∩ B)). Mathematically, this can be expressed as:
P(A ∪ B) = P(A) + P(B) - 2P(A ∩ B)
Substituting the given values into this equation, we get:
0.8 = P(A) + P(B) - 2P(A ∩ B)
Now, we are given that P(A given B) = 0.6. This means that the probability of A occurring, given that B has already occurred, is 0.6. In other words:
P(A | B) = 0.6
Using the definition of conditional probability, we can express P(A ∩ B) as:
P(A ∩ B) = P(B) * P(A | B)
Substituting the given value, we get:
P(A ∩ B) = P(B) * 0.6
Now, we can rewrite the equation for P(A ∪ B) as:
0.8 = P(A) + P(B) - 2P(B) * 0.6
Simplifying this equation, we get:
0.8 = P(A) + P(B) - 1.2P(B)
Rearranging terms, we have:
P(A) = 0.8 + 1.2P(B) - P(B)
Combining like terms, we get:
P(A) = 0.8 + 0.2P(B)
Since we don't have the value of P(B) directly, we need to find a relationship between P(B) and P(A ⊕ B). Recall that P(A ⊕ B) is given as 0.6, and we can express it as:
P(A ⊕ B) = 1 - P(A ∩ B)
Substituting the earlier definition, we have:
0.6 = 1 - P(B) * P(A | B)
Simplifying, we get:
0.6 = 1 - P(B) * 0.6
Rearranging terms, we have:
P(B) * 0.6 = 1 - 0.6
P(B) * 0.6 = 0.4
Dividing both sides by 0.6, we get:
P(B) = 0.4 / 0.6
Simplifying further, we obtain:
P(B) = 2/3
Finally, substituting this value back into the equation for P(A), we have:
P(A) = 0.8 + 0.2 * (2/3)
Simplifying, we get:
P(A) = 0.8 + 0.4/3
Finding a common denominator, we have:
P(A) = (24/30) + (4/30)
Combining fractions, we get:
P(A) = 28/30
Simplifying further, we obtain the final answer:
P(A) = 14/15
Therefore, P(A) is equal to 14/15 or approximately 0.933.