Conditional probability and the product rule
If one is planning a picnic for the Fourth of July, one does not care what fraction of the days in the year it rains, but what fraction of the days in July it rains. For example, the probability of getting the jack of spades is 1/52, but if you know you are getting a black card, the probability becomes 1/26, if you know you are getting a jack the probability is 1/4, and if you know you are geting a black jack the probability is 1/2. Similarly, if you know you are getting a red card, the probability of getting the jack of spades is zero.
Formally we define the probability of A conditioned on B as P(A|B) = P(A and B)/P(B). The division on the right hand side assures that conditional probabilities sum to one as well as unconditional probilities. Especially with equally likely events, conditional probabilities can be interpreted as probabilities in a restricted universe. Hence the probability of getting the queen of spades conditioned on (or given that) you get a spade is P(Q and S)/P(S) = (1/52)/(1/4) = 1/13 by the formula, but can also be calculated as 1/13 as equally likely events in the universe restricted to spades.
Note that in general P(A|B) is not equal to P(B|A)
Exercise: P(heart|jack)=? P(jack|heart)=? P(one-eyed jack|heart)=? P(king|face card)=? P(face card|king)=?
From the definition of conditional probability, it is immediate that P(A and B) = P(A|B)P(B) = P(B|A)P(A). This is the product rule, e.g., P(king|heart) = 1/13, P(heart) = 1/4, therefore P(king and heart) = 1/13 × 1/4 = 1/52
Definition: A and B are said to be independent if P(A|B) = P(A); this means that conditioning on B gives you no further information. For example, knowing that one is a boy provides no further information as to whether one will get an A. Substituting this definiton into the product rule yields an alternative definition of independence: A and B are independent if P(A and B) = P(A) × P(B). One can readily verify that being a heart and being a jack are independent, but being a one-eyed jack and being a heart are not independent.
Competencies: If you roll a pair of dice, what is the probability that one die is a 5 if the total is 8? What is the probability that the total is 8 if one die is a five? Are one die being a 5 and the total being 8 independent events?
Reflection: What, if any, are the relationships between complementary, mutually exclusive, and independent.
Challenge:If P(A) = .4, P(B) = .5, and P(A or B) = .7, are A and B independent?
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