Greatest Probability Neither Event Will Happen
Date: 02/02/2002 at 12:43:56 From: Lucia Subject: Probability Dear Dr.Math, Here is a GMAT question that I couldn't answer. The probability that event A will happen is 0.54, and the probability that event B will happen is 0.68; what is the greatest probability that neither of the events will happen? I subtracted the probability of each event from 1, to find the probability that each event would not happen; and then multiplied those probabilities. However, I didn't match any of the answer choices. For instance: P (A) not happens: 1-.54 =.46 P (B) not happens: 1-.68= .32 .46*.32 = 0.1472 Could you please help me understand the reasoning of this question. Thanks a lot. Lucia
Date: 02/04/2002 at 17:11:39 From: Doctor Douglas Subject: Re: Probability Hi, Lucia Thanks for submitting your question to the Math Forum! The question does not say anything about the independence of the two events, so we are not allowed to multiply probabilities, as you have done. In this case, we have to try to find the probability of the event X = "neither A nor B", and see what values it can take. In particular, what is the largest value of Pr(X)? Let's imagine a stick of length 1, and let this represent probability. ************************* Now, from the problem, 46% of the stick is 'not A' (denote this event by a lowercase a), and 54% of the stick is 'A', and this is represented by the following picture. Note that we don't know where on the stick is 'A' and where is 'a'. aaaAAAaaAAAaAaaAAAAAAaaAA Similarly, Pr(B)=.68, and the stick can also be represented by BBBBBBBBBBBBBBBBBbbbbbbbb .....................XX.. note event X is (a and b), In this case, only a small sliver of the stick near the end is composed of b and a simultaneously (i.e., X). If we could rearrange the various letters around so that there were more parts of the stick that had both lowercase letters, we could make the probability of X be larger. In other words, it is possible for the stick to look like this: AAAAAAAAAAAAAAAaaaaaaaaaa BBBBBBBBBBBBBBBBBbbbbbbbb .................XXXXXXXX so here we see that the greatest that Pr(X) can be is when all of the outcomes 'b' are also characterized by the outcome 'a'. Whichever of Pr(a) or Pr(b) is smaller will determine the entire size of this overlap region X. In the problem that you were given, Pr(b) = 0.32 is the smaller of Pr(a) and Pr(b), so that the maximum possible value of Pr(X) is 0.32. This is not so much a rule as the sort of logical reasoning that we employ in manipulating numbers and sets of numbers: example: if x+y = 5 and both x and y are nonnegative, what is the largest possible value for x? answer: the bigger x is, the smaller y is, since they sum to 5. the smallest y can be is 0, since otherwise it would be negative, which is a contradiction. Thus the greatest x can be is 5, in the case that y=0. Note that this reasoning doesn't say that x is in fact 5, only that the greatest possible value for x is 5. Sometimes we write this fact as x <= 5. I hope this helps. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
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