Conditional ProbabilityDate: 12/31/97 at 14:21:01 From: chris Bonnette Subject: Conditional probability Thank you for providing such an interesting way of finding out math information. It is much appreciated. I tutor probability and statistics and people always seem to be very confused by conditional probability: P(AIB) = P(both events)/P(known event occurred) = P(AB)/P(B) The confusion seems to lie in the use of the multiplication rule. If the events are dependent, they seem to feel that they end up in an infinite loop: P(AB) = P(A)*P(BlA)...but P(BlA) = P(BA)/P(A). It appears that everything just cancels and you are left with P(AB) and are no closer to finding the solution. I'm looking for a clear and nonconfusing way to initially teach this concept to them so that this confusion doesn't even start. Any suggestions? Chris Date: 12/31/97 at 17:21:18 From: Doctor Anthony Subject: Re: Conditional probability A picture of the sample space allows you to calculate conditional probabilities without too much of the confusing notation. Example: A bag contains 5 similar coins except that one is double-headed. A coin is chosen at random and is tossed 4 times. Each time it lands heads. What is the probability that the double-headed coin was chosen? Chosen coin is Chosen coin is double-headed. normal. prob = 1/5 prob = 4/5 ---------------------------------------------------- (1/5) x 1 (4/5) x (1/2)^4 4 heads obtained. ---------------------------------------------------- xxxxxx xxxxxxxxxxx other result. ----------------------------------------------------- The sample space is restricted to the top line since we are told that 4 heads were obtained. We can find the probability that we are in the lefthand box by putting its value over the total value of the top line. (1/5) Prob.double-headed coin = -------------------- (1/5) + (4/5)(1/16) 1 = ----------------- 1 + 1/4 4 = -------- 4 + 1 = 4/5 So the probability that we have the double-headed coin, on the evidence of throwing 4 heads in a row, is 4/5. Of course, even one tail would have made it certain that we had a normal coin. This method allows you to understand what is going on without the somewhat opaque notation of conditional probability. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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