Rules on probability
Answers
Step-by-step explanation:
Rule of Subtraction
The probability of an event ranges from 0 to 1.
The sum of probabilities of all possible events equals 1.
The rule of subtraction follows directly from these properties.
Rule of Subtraction. The probability that event A will occur is equal to 1 minus the probability that event A will not occur.
P(A) = 1 - P(A')
Suppose, for example, the probability that Bill will graduate from college is 0.80. What is the probability that Bill will not graduate from college? Based on the rule of subtraction, the probability that Bill will not graduate is 1.00 - 0.80 or 0.20.
Rule of Multiplication
The rule of multiplication applies to the situation when we want to know the probability of the intersection of two events; that is, we want to know the probability that two events (Event A and Event B) both occur.
Rule of Multiplication The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred.
P(A ∩ B) = P(A) P(B|A)
Example
An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?
Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is black. We know the following:
In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10) * (3/9) = 12/90 = 2/15 = 0.133
Rule of Addition
The rule of addition applies to the following situation. We have two events, and we want to know the probability that either event occurs.
Rule of Addition The probability that Event A or Event B occurs is equal to the probability that Event A occurs plus the probability that Event B occurs minus the probability that both Events A and B occur.
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Note: Invoking the fact that P(A ∩ B) = P( A )P( B | A ), the Addition Rule can also be expressed as:
P(A ∪ B) = P(A) + P(B) - P(A)P( B | A )
Example
A student goes to the library. The probability that she checks out (a) a work of fiction is 0.40, (b) a work of non-fiction is 0.30, and (c) both fiction and non-fiction is 0.20. What is the probability that the student checks out a work of fiction, non-fiction, or both?
Solution: Let F = the event that the student checks out fiction; and let N = the event that the student checks out non-fiction. Then, based on the rule of addition:
P(F ∪ N) = P(F) + P(N) - P(F ∩ N)
P(F ∪ N) = 0.40 + 0.30 - 0.20 = 0.50
Answer:
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