vabzgupta237 wrote:
AnthonyRitz wrote:
vabzgupta237 wrote:
I also faced the same question in
veritasprep test, and didnt understood the explaination of
veritas prep can someone help please. Have GMAT in 7 Days.
Hey guys. How can I help? What part of the solution are you having trouble with?
The whole part, how to start following the basic knowledge, not a specialised approach for this question.
Okay, let me try to walk you through it.
First of all, whenever you're doing a probability problem, you always have at least two possible approaches. One is the direct approach -- calculate the good outcomes. The other is the complementary approach -- calculate the bad outcomes, and then subtract from 1. Either approach will work here, but in cases involving "at least one" (at least one pair of matching socks, etc.) it is often best to go with the complementary approach. The opposite of "at least one" is "none,"and "none" is usually less difficult to calculate than all the different ways of getting "at least one."
Okay, so two approaches. Bradley did a solid job of laying out a direct approach. Pick socks one at a time.
Is the first one a pair? Definitely not.
Is the second one a pair? Only if it matches the first one -- that's one good option out of eleven total, 1/11.
Is the third one a pair? To make it this far, we already are in the case of no pair yet, which is 10/11, and then we are holding two mismatched socks. We will pair one of them 2/10 of the time, so our chance of a good outcome is now 1/11 + 10/11 * 2/10 = 3/11. Note that to get this expression I have applied the Basic Counting Principle -- "and" means "multiply"; "or" means "add." This rule is critical for any problem involving counting. That means almost all combinatorics and much of probability needs this rule.
Is the fourth one a pair? To make it this far, we still have no pair, which happens 8/11 of the time. Of the nine remaining socks, three will match something we have, so the probability is 3/11 + 8/11 * 3/9 = 27/99 + 24/99 = 51/99, which is more than 50%. So four socks is the answer.
The complementary approach focuses on not making a pair.
The first sock avoids a pair 100% of the time.
The next sock avoids a pair 10/11 of the time (everything but the mate of the first sock pulled is fine).
The next sock avoids a pair 8/10 of the time (ten socks are left, but two of them pair the two already pulled). Overall, we have no pair 100% * 10/11 * 8/10 = 8/11 of the time. So we have a pair 1 - 8/11 = 3/11 of the time (note: same result we got the other way).
The next sock avoids a pair 6/9 of the time (note that every step there is one less sock in the drawer but two less *good* socks in the drawer -- not only the one we picked but also its mate will now fail to avoid a pair). Overall, we have no pair 100% * 10/11 * 8/10 * 6/9 = 48/99 of the time. So we have a pair 1 - 48/99 = 51/99 of the time (again, same result with either approach). Four socks is still the right answer.
I want to be clear that these are not fringe or specialized approaches. Complementary probability is a core, fundamental tool for GMAT probability questions. At the 700 level (where this question is squarely located), you often won't have much chance without it. In fact, this problem follows a long line of very similar problems that use these approaches. With practice, they can be done correctly in two minutes or less.
I'm happy to explain further and answer any questions you may have.
I hope this helps!!