In a department store prize box, 40% of the notes give...

Hello mahakmalik,

How it was solved above is from a formula specifically for the probability when x will occur and when not x will occur, the Bernoulli formula. The initial part of your expression selects three out of five people correctly. However, the second part of the expression is essentially the same thing except for the probability for not x, which is an equally important part as anything else. Thus, extend the expression with the probability of not x:

\(5c3 * ((\frac{8c1*8c1*8c1)}{20c1*20c1*20c1})*(\frac{12c1*12c1}{20c1*20c1}))\)

Then, you have a identical formula but with bigger numbers, which leads to the same answer. Hope this helps!

Kr,
Mejia

ok, first, I did not how to tackle this question...
so we we need to have 3 winners out of 5. this can be done in 5C3 ways, or 10 ways.
WWWLL
LWWWL
LLWWW
etc. so 10 ways.

now, the probability of winning is 4/10, and probability of loosing is 6/10
i did not simplify, since working with 10's are much easier.

so 4/10 must be 3 times, and 6/10 twice.
(4/10)^3*(6/10)^2 * 10 (10 ways we can arrange winners and losers).
ok, so
(4/10)^3 = 64/1000
(6/10)^2 = 36/100
now, the first one can be multiplied by 10, and rewritten as 64/100
now, multiply 64/100 with 36/100. this will be equal to 2304/10000
this is aprox. 23/100 or 0.23

I just have one question regarding this,
3 out of 5 people that draw one after the other, doesnt it suggest that the 3 people should be consucutive?

like:
5 people : [Draw,Draw,Draw,Not Draw,Not Draw] or [Not Draw,Draw,Draw,Draw,Not Draw] or [Not Draw,Not Draw,Draw,Draw,Draw]?


Any people who consider this a valid doubt , please consider Kudos.

Why do we have to divide by 3!2!? I thought that simply multiply it by 5! as it is permutation.

because 3 winners can be arranged in different orders, and it doesn't matter how they are arranged.
it might be W1W2W3 or W2W1W3 -> note that winner is a winner, and they are all the same. because of this, we have to divide by 3! since we have 2 non-winners, those one can be arranged as well in 2! ways, since we do not distinguish between L1 and L2.

Hi Zurich,

In this case the order of the people selecting the notes does not matter. It can be looked at as a combination, instead of a permutation. The number of ways to select 3 winners out of 5 is given by 5C3 = \(\frac{5!}{3!2!}\)

It is mathematically the same as saying we will find the number of permutations of 5 people, but when there are 3 identical and 2 identical people in the group. In this case, we must divide the permutation of 5 people, 5!, by the number of arrangements of each of the identical elements, 3! and 2!.

This again equals \(\frac{5!}{3!2!}\)

The only difference is the way we explain it. Either we are choosing 3 out of 5 where the order doesn't matter, or we are arranging 3 identical and 2 identical items in a row. It works out to the same thing.

I hope that helps to clear it up!

Can someone please explain how does the probability of winning remains the same in all 3 cases? Lets say that there are 100 notes, 40 are winning notes.
P1 picks first note: probability of winning = 0.4
P2 picks second : P(W) = 39/99 = 0.39
P3 picks third: P(W) = 38/98 = 0.387

Hence, without knowing the number of tickets, we would not be able to find the probability of winning! Are we assuming that the tickets are by replacement? This would not make sense

The wording of the question is awful, so I'd suggest just ignoring it altogether, but when the question says each person will "immediately return their note into the box", the question is trying to say that the selections are made with replacement, so the probability of winning is always 2/5. You're entirely correct that without this information, the total number of "notes" affects the answer.

The question is misusing nouns (no one would ever use the word "notes" in this way), the writer in me cringes at the phrase "dreamy vacation", and when the question asks for the probability that "3 out of 5 people that draw the notes one after the other... get a dreamy vacation?" it is not at all clear what the question is even asking. Do we simply want three of the five people to win? Or do the three winners need to occur "one after the other"? I'm not sure what the source is, but no real GMAT question would ever be worded in a remotely similar way.