Solution


Here we have an organization of 700 members. Now there are 120 lawyers in the 700 members thus there is 580 not lawyers. Now we have to choose 2 not lawyers.

So the probability becomes \(\frac{580}{700}*\frac{579}{699}\).

Now we can use the calculator available at the GRE to find the value and it is close to 0.7.

Note that the second term is \(\frac{(580-1)}{(700-1)}\) since one not lawyer was already selected in the first attempt.
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A student asked me to solve this question (and answer a few other questions). So, here it goes...

SOLUTION #1: Probability rules

Question: Are we selecting people with replacement or without replacement?
We're told that two members are selected, so we can assume that the two members are different.
In other words, we're selecting people without replacement.
ASIDE: If this were my question, I would have written "two DIFFERENT members will be selected..." to be free from ambiguity. That said, this is an official question, so this is what we have to go with.

Of the 700 members of a certain organization, 120 are lawyers.
So we have 120 lawyers
And we have 580 non-lawyers

Two members of the organization will be selected at random. Which of the following is closest to the probability that neither of the members selected will be a lawyer?
P(neither selection is a lawyer) = P(1st selection is non-lawyer AND 2nd selection is non-lawyer)
= P(1st selection is non-lawyer) x P(2nd selection is non-lawyer)
= 580/700 x 579/699
≈ 0.69

Answer: C

ASIDE: I should also mention that, if we were to treat this question as a replacement question, we'd still get the same answer, since the value of 579/699 is very close to the value of 580/700

Cheers,
Brent
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SOLUTION #2: Counting methods

Of the 700 members of a certain organization, 120 are lawyers.
So we have 120 lawyers
And we have 580 non-lawyers

P(neither selection is a lawyer) = (number of ways to select 2 non-lawyers)/(TOTAL number of ways to select two members)

number of ways to select 2 non-lawyers
There are 580 non-lawyers, and we want to select 2 of them.
Since the order in which we select the 2 non-lawyers does not matter, we can use combinations.
580C2 = (580)(579)/2! = (290)(579)

TOTAL number of ways to select two members
There are 700 members in total, and we want to select 2 of them
Since the order in which we select the 2 members does not matter, we can use combinations.
700C2 = (700)(699)/2! =(350)(699)


we get; P(neither selection is a lawyer) = (290)(579)/(350)(699)
≈ 0.69

Answer: C

Cheers,
Brent
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Can we do this by using this formula?
(1- favorable outcome/total outcome)
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You COULD use the complement formula, however this will require a lot more work.

We want P(neither selected person as a lawyer)

The complement of neither selected person as a lawyer is:
1st selection is a lawyer and 2nd selection is not a lawyer.
OR
1st selection is not a lawyer and 2nd selection is not a lawyer.
OR
1st selection is a lawyer and 2nd selection is a lawyer.

So, to apply the complement formula, we must calculate all three probabilities.
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We want a case where we choose two people and they are both NOT lawyers.

So, p(A)*p(B)

= \(\frac{580}{600} * \frac{579}{599}\) (we reduce 1 lawyer from the second expression since we already chose one in the first)
= 0.68

0.7 is closest to our answer.

Therefore, C is the right choice

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