If P(A)=.3 and P(B)=.45 find… P(A or B) if A and B are 1)mutually exclusive & 2)independent.
Also, I need help understanding what independent and mutually exclusive means and what that does.
Either-or probabilities are found by adding the individual probabilities.
http://math.stackexchange.com/questions/941150/what-is-the-difference-between-independent-and-mutually-exclusive-events
To find P(A or B), we need to first understand the concepts of mutually exclusive and independent events.
1) Mutually exclusive events: Two events, A and B, are said to be mutually exclusive if they cannot both occur at the same time. In other words, if event A happens, event B cannot happen, and vice versa.
2) Independent events: Two events, A and B, are said to be independent if the occurrence of one event does not affect the probability of the other event occurring. In other words, whether event A happens or not, it does not change the probability of event B happening, and vice versa.
Now, let's calculate P(A or B) for the given probabilities:
1) Mutually exclusive events:
When events A and B are mutually exclusive, we can use the addition rule of probability. This rule states that the probability of the union of two mutually exclusive events is equal to the sum of their individual probabilities.
P(A or B) = P(A) + P(B)
Given P(A) = 0.3 and P(B) = 0.45:
P(A or B) = 0.3 + 0.45
P(A or B) = 0.75
2) Independent events:
When events A and B are independent, we can use the multiplication rule of probability. This rule states that the probability of the intersection of two independent events is equal to the product of their individual probabilities.
P(A and B) = P(A) * P(B)
Given P(A) = 0.3 and P(B) = 0.45:
P(A and B) = 0.3 * 0.45
P(A and B) = 0.135
In this case, we are looking for P(A or B), which is the probability that either event A or event B (or both) occurs. To find it, we can use the formula for the union of two events:
P(A or B) = P(A) + P(B) - P(A and B)
Using the probabilities given above:
P(A or B) = 0.3 + 0.45 - 0.135
P(A or B) = 0.615
To find P(A or B) when A and B are mutually exclusive, you simply add the individual probabilities of A and B.
P(A or B) = P(A) + P(B) = 0.3 + 0.45 = 0.75
So, if A and B are mutually exclusive, the probability of A or B occurring is 0.75.
Now, let’s discuss what mutually exclusive and independent mean.
1) Mutually Exclusive: When two events are mutually exclusive, it means that they cannot happen at the same time. If event A occurs, then event B cannot occur, and vice versa. In other words, there is no overlap between the occurrences of A and B. For example, if you flip a coin, getting a heads and getting a tails are mutually exclusive events.
2) Independent: When two events are independent, it means that the occurrence or non-occurrence of one event has no effect on the occurrence of the other event. In other words, the probability of event A happening is the same irrespective of whether event B happened or not, and vice versa. For example, if you roll a dice, getting an odd number and getting a number greater than 3 are independent events.
Now, to find P(A or B) when A and B are independent, we need to use the formula:
P(A or B) = P(A) + P(B) - P(A and B)
However, since A and B are independent, P(A and B) will be simply the product of P(A) and P(B).
P(A or B) = P(A) + P(B) - P(A) * P(B) = 0.3 + 0.45 - (0.3 * 0.45) = 0.75 - 0.135 = 0.615
So, if A and B are independent, the probability of A or B occurring is 0.615.