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'll need to use the given information and apply some basic probability concepts.
Let's represent the events as A and B. From the given information, we have:
P(A given B) = 0.6
P(at least one of the two events occurs) = 0.8
P(exactly one of the two events occurs) = 0.6
First, let's consider the concept of "at least one of the two events occurs." This means we need to find the probability of A occurring, B occurring, or both A and B occurring. In probability notation, this can be represented as:
P(A or B) = P(A) + P(B) - P(A and B)
Using this concept, we can rewrite the given information:
0.8 = P(A) + P(B) - P(A and B) (equation 1)
Now, let's consider the concept of "exactly one of the two events occurs." This means we need to find the probability of A occurring and B not occurring, or B occurring and A not occurring. In probability notation, this can be represented as:
P((A and not B) or (B and not A)) = (P(A) * (1 - P(B))) + (P(B) * (1 - P(A)))
Using this concept, we can rewrite the given information:
0.6 = (P(A) * (1 - P(B))) + (P(B) * (1 - P(A))) (equation 2)
Now we have a system of two equations (equation 1 and equation 2) with two unknowns (P(A) and P(B)). We can solve this system of equations to find the value of P(A).
Let's solve the equations:
From equation 2, simplify the expression:
0.6 = P(A) - P(A) * P(B) + P(B) - P(A) * P(B)
Combine like terms:
0.6 = P(A) + P(B) - 2P(A) * P(B)
Rearrange the equation:
0.6 - P(A) - P(B) = -2P(A) * P(B)
Multiply both sides by -1:
P(A) + P(B) - 0.6 = 2P(A) * P(B)
Now substitute this equation back into equation 1:
0.8 = P(A) + P(B) - P(A and B)
Replace P(A) + P(B) in equation 1 with 0.8 + 0.6 - 0.6 = 0.8:
0.8 = 0.8 + 2P(A) * P(B) - P(A and B)
Rearrange the equation:
0 = 2P(A) * P(B) - P(A and B)
Now we can see that the expression on the right side of the equation must be zero:
2P(A) * P(B) - P(A and B) = 0
Since the probability of an event cannot be negative, we have two possibilities:
1. P(A) = 0
2. P(B) = 0
However, we're given that at least one of the two events occurs, so neither P(A) nor P(B) can be zero.
Therefore, we can conclude that P(A) must not be equal to zero. Thus, P(A) = 2/3.
Therefore, the value of P(A) in this scenario is 2/3.