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/