By William M. Briggs
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Hint The probability that A is true is one minus the probability that A is false. List one situation where you have an event or observation statement A and another B such that the probability of A being true is irrelevant to knowledge that B is true. That is, that Pr(A|BE) Pr(B|E) = Pr(A|E) Pr(B|E). Write out your A, B, and E carefully. Most gambling situations fit this scenario well. List one situation where you have an event or observation statement A and another B such that the probability of A being true is relevant to knowledge that B is true.
24. 001. 0 5. HOMEWORK 31 where you must remember that a0 = 1 (for any a you will come across, even a = 0). You see the pattern by now. This means we have another formula to add to our collection. This one is called the binomial and it looks like this: n k (14) Pr(k|n, p, EB ) = p (1 − p)n−k . k There is a subtle shift in notation with this formula, made to conform with tradition. ” For general situations, k is the number of “successes”: or, K = “The number of successes is k”. Everything to the right of the “|” is still information that we know.
Incidentally, if C were also true, then Pr(A|E) = Pr(ABC|E), too. We can prove this by recourse to rule number 3. Pr(AB|E) = Pr(A|BE) Pr(B|E) = Pr(A|BE) = Pr(A|E) because Pr(B|E) = 1 and BE=E (remembering the Boolean algebra rules). 4. PROBABILITY RULE NUMBER 4: BAYES’S RULE 21 We can now use the simple consequence of logic that A=AB to help us calculate the probability of A using something called total probability: (5) Pr(A|E) = = = = = Pr(AB|E) Pr(A(B1 or B2 or . . or Bn )|E) Pr(AB1 or AB2 or .
Breaking the Law of Averages: Real-Life Probability and Statistics in Plain English by William M. Briggs