Suppose that A and B are events defined on a common sample space and that the following probabilities are known. Find P(A or B). (Give your answer correct to two decimal places.)
P(A) = 0.34, P(B) = 0.41, P(A | B) = 0.2
Well, well, well, we have a probability problem here. So, we want to find the probability of either A or B happening.
Now, we can start by thinking about the probability of just A happening. We know that P(A) = 0.34.
Then, we can think about the probability of just B happening. We know that P(B) = 0.41.
But, we have to consider that A and B might not be mutually exclusive. In other words, they could both happen at the same time. That's where P(A | B) comes in. P(A | B) is the probability of A happening, given that B has already happened. In this case, it's 0.2.
So, to find P(A or B), we have to consider the overlap between A and B. The formula for that is:
P(A or B) = P(A) + P(B) - P(A and B)
Now, let's plug in the numbers:
P(A or B) = 0.34 + 0.41 - P(A and B)
But wait! We need to find P(A and B) first. The formula for that is:
P(A and B) = P(A | B) * P(B)
Let's calculate that:
P(A and B) = 0.2 * 0.41 = 0.082
Now, let's go back to the formula for P(A or B) and plug in the numbers again:
P(A or B) = 0.34 + 0.41 - 0.082
By doing the math, we find:
P(A or B) = 0.668
So, the probability of A or B happening is 0.67 (rounded to two decimal places).
To find the probability of "A or B", denoted as P(A or B), we can use the formula:
P(A or B) = P(A) + P(B) - P(A and B)
Given the information, we have:
P(A) = 0.34
P(B) = 0.41
P(A | B) = 0.2 (probability of A given B)
We don't have the direct value of P(A and B), which represents the probability of both A and B happening at the same time. However, we can find it using the formula:
P(A and B) = P(B) * P(A | B)
Substituting the values, we have:
P(A and B) = 0.41 * 0.2 = 0.082
Now we can calculate P(A or B):
P(A or B) = P(A) + P(B) - P(A and B)
= 0.34 + 0.41 - 0.082
= 0.668
Therefore, P(A or B) is approximately 0.67.