Conditional probability
Probability of an event given that another event has already occurred
Conditional probability is the probability of an event occurring given that another event has occurred. It is represented by P(A|B), where A and B are events and P(A|B) represents the probability of A given that B has occurred.
To calculate conditional probability, we use the formula:
P(A|B) = P(A and B)/P(B)
where P(A and B) represents the probability of both A and B occurring, and P(B) represents the probability of B occurring.
For example, let’s say we are interested in the probability of a student getting an A in a class, given that they attended 90% of the lectures. Let A be the event of getting an A and B be the event of attending 90% of the lectures. Let’s assume that the probability of getting an A is 0.20 and the probability of attending 90% of the lectures is 0.50.
To calculate P(A|B), we need to first calculate P(A and B). Since these events are not independent, we cannot simply multiply the probabilities. Instead, we need to use the formula:
P(A and B) = P(A)*P(B|A)
where P(B|A) represents the probability of attending 90% of the lectures given that the student got an A. Let’s assume that this probability is 0.80.
Now we can calculate P(A and B) = 0.20*0.80 = 0.16.
Next, we can calculate P(B) by using the formula:
P(B) = P(A)*P(B|A) + P(not A)*P(B|not A)
where P(not A) represents the probability of not getting an A. Let’s assume that P(not A) = 0.80 and P(B|not A) = 0.30.
Now we can calculate P(B) = 0.20*0.80 + 0.80*0.30 = 0.44.
Finally, we can calculate P(A|B) by using the formula:
P(A|B) = P(A and B)/P(B) = 0.16/0.44 = 0.36.
Therefore, the probability of a student getting an A in the class given that they attended 90% of the lectures is 0.36.
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