Suppose that A and B are independent events such thatP(A)=.50 andP(B)=.30 .
Find and P(A U B) and P(A upside down U B) .
P(A or B) but not both
= P(A) + P(B) - P(both A and B)
P(A or B) = P(A U B)
P intersection which is both at once and your upside down U is
P(A)P(B) = .5*.3 = .15
so
P(A U B) = .5 + .3 - .5 = .65
P(A or B)
= P(A) + P(B) - P(both A and B)
P(A or B) = P(A U B)
P intersection which is both at once and your upside down U is
P(A)P(B) = .5*.3 = .15
so
P(A U B) = .5 + .3 - .15 = .65
To find P(A U B) and P(A upside down U B), we need to know if A and B are mutually exclusive or not.
If A and B are mutually exclusive (i.e., they cannot occur at the same time), then P(A U B) = P(A) + P(B). However, since A and B are independent in this case, they can occur at the same time, so A and B are not mutually exclusive.
If A and B are not mutually exclusive (i.e., they can occur at the same time), then P(A U B) = P(A) + P(B) - P(A ∩ B).
Given that A and B are independent events, P(A ∩ B) = P(A) * P(B) = 0.50 * 0.30 = 0.15.
Therefore, P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.50 + 0.30 - 0.15 = 0.65.
For P(A upside down U B), this is the probability that either A or B occurs, but not both at the same time. Since A and B are independent events, they can occur at the same time, so P(A upside down U B) = P(A) + P(B) - 2 * P(A ∩ B) = 0.50 + 0.30 - 2 * 0.15 = 0.50.
To find the union (U) of two events, you need to determine the probability that either event A or event B (or both) occur. This can be found by adding the individual probabilities of A and B and subtracting the probability that both events occur (intersection). The upside-down U symbol (∩) represents the intersection of two events, which is the probability that both event A and event B occur simultaneously.
Given:
P(A) = 0.50
P(B) = 0.30
To find P(A U B):
P(A U B) = P(A) + P(B) - P(A ∩ B)
Since the events A and B are given to be independent:
P(A ∩ B) = P(A) * P(B)
Substituting the values:
P(A U B) = P(A) + P(B) - P(A) * P(B)
P(A U B) = 0.50 + 0.30 - (0.50 * 0.30)
P(A U B) = 0.80 - 0.15
P(A U B) = 0.65
Therefore, the probability of event A or event B (or both) occurring is 0.65.
To find P(A upside down U B):
As mentioned earlier, the upside-down U symbol (∩) represents the intersection of events A and B.
P(A upside down U B) = P(A ∩ B)
Since the events A and B are given to be independent:
P(A ∩ B) = P(A) * P(B)
Substituting the values:
P(A upside down U B) = P(A) * P(B)
P(A upside down U B) = 0.50 * 0.30
P(A upside down U B) = 0.15
Therefore, the probability of event A and event B occurring simultaneously is 0.15.