probability

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probability

by champmag » Wed May 18, 2011 9:24 am
Out of 100 lottery tickets, each having 50% chance of winning, 2 tickets are bought at random. What is the probability that at least one ticket will win?

A.1/4
B.1/2
C.49/198
D.149/198
E.3/4
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by pemdas » Wed May 18, 2011 11:18 am
the essence of ANY probability is an experiment. If we replicate our experiment - draw a lottery - one hundred times we should have 50 win or loose lotteries, because the probability that each will win/loose is 50%. But on the second draw we have 49/99 for either win or loose. Suppose 50/100 is win and we selected winning lottery on the first draw, and now we have 49 winning lotteries in the bag. We still have 50 loser lotteries, so the probability we select the second lottery and it looses is 50/99. BUT the second probability that it doesn't loose and win is 49/99. Or vice versa all two lose -> 1/2*49/99. If we subtract this probability from 1 we get the required probability of at least two win (one+two) 1-49/198=149/198
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by Stuart@KaplanGMAT » Wed May 18, 2011 12:03 pm
champmag wrote:Out of 100 lottery tickets, each having 50% chance of winning, 2 tickets are bought at random. What is the probability that at least one ticket will win?

A.1/4
B.1/2
C.49/198
D.149/198
E.3/4
Hi,

since each ticket has a 50% chance of winning, we must treat each as an independent event; nowhere does the question suggest otherwise.

Since there's a 50% chance of winning (and, accordingly, a 50% chance of losing), we can treat this just as we would the following coin flip problem:
If you flip a fair coin twice, what's the probability of getting at least one head?
Like most multiple scenario probability questions, we can take one of two general approaches:

1) calculate the probability of each scenario we do want, then add them together; or
2) calculate the probability of the scenario(s) that we don't want, then subtract that from 100%.

On the GMAT, method (2) is almost always quicker.

For this question, the only thing we don't want is two tails (or two losses, using the original language).

The probability of tails-tails is 1/2 * 1/2 = 1/4.

Accordingly, the probability of NOT tails-tails is:

1 - 1/4 = 3/4

Solving it the first (and longer) way, there are three scenarios that match "at least one" head:

H-H
H-T
T-H

whose probabilities are:

1/2 * 1/2 = 1/4
1/2 * 1/2 = 1/4
1/2 * 1/2 = 1/4

So the probability of getting HH or HT or TH = 1/4 + 1/4 + 1/4 = 3/4

Choose E!
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by pemdas » Wed May 18, 2011 12:56 pm
@Stuart, i understand your logic, BUT please you also try ONE only ONE lottery in two, three or more different experiments and you will see that the events are not independent BUT mutually exclusive. Yes it can be either win or loose - mutually exclusive, BUT this does not mean independence yet. Independence is when occurrence of one event does not change the occurrence of another event. If you toss the fair coin - Yes, the same coin can land Tail or Head each time, BUT if you draw ONE lottery it will be either win or loose for ever. The occurrence of win and loose will not switch like T-H. The events win-loose are mutually exclusive BUT not independent.
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by Stuart@KaplanGMAT » Wed May 18, 2011 1:14 pm
pemdas wrote:@Stuart, i understand your logic, BUT please you also try ONE only ONE lottery in two, three or more different experiments and you will see that the events are not independent BUT mutually exclusive. Yes it can be either win or loose - mutually exclusive, BUT this does not mean independence yet. Independence is when occurrence of one event does not change the occurrence of another event. If you toss the fair coin - Yes, the same coin can land Tail or Head each time, BUT if you draw ONE lottery it will be either win or loose for ever. The occurrence of win and loose will not switch like T-H. The events win-loose are mutually exclusive BUT not independent.
Hi,

the problem is that you're making assumptions based on information that you don't have - a practice that will get you in a lot of trouble during the GMAT.

Here are some things we don't know:

are all 100 tickets for the same lottery, or is each one for a separate lottery;

is there a fixed or unlimited number of prizes; and

once one ticket wins, does that diminish the probability of the next ticket winning?

You've assumed answers to all of these questions that aren't in evidence in the stem.

Here's how you've interpreted the question:
In a certain lottery, 100 tickets were sold and there are 50 prizes available. If each ticket has an equal chance to win and can only win one prize, then if two tickets are drawn at random what's the probability that at least one of the two of them wins a prize?
However, that's very different than the question provided.
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by pemdas » Wed May 18, 2011 1:17 pm
i would ask poster of original question to reveal the source and official explanation, if possible.
thanks
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by pemdas » Wed May 18, 2011 1:28 pm
@Stuart, isn't that you are making assumptions in the previous post
are all 100 tickets for the same lottery, or is each one for a separate lottery;

is there a fixed or unlimited number of prizes; and

once one ticket wins, does that diminish the probability of the next ticket winning?
assumption No 1: each lottery is for separate lottery?
assumption No 2: each lottery is attached binomial distribution success-failure rates 50%?

@champmag: what's the source? any official explanation in your book?
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by Ian Stewart » Wed May 18, 2011 8:07 pm
I'd interpret the question the same way as Stuart did above, since the wording strongly suggests each ticket has a 50% probability of winning, independent of the results of any other ticket. But the intention of the question is unclear - why do they mention 100 tickets at all if each ticket is independent of the others? It's just a badly written question, so there isn't much point in debating how to interpret it.
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by manpsingh87 » Wed May 18, 2011 8:54 pm
Ian Stewart wrote:I'd interpret the question the same way as Stuart did above, since the wording strongly suggests each ticket has a 50% probability of winning, independent of the results of any other ticket. But the intention of the question is unclear - why do they mention 100 tickets at all if each ticket is independent of the others? It's just a badly written question, so there isn't much point in debating how to interpret it.
i completely agree with both ian and stuart,, i also solve the question in the same way as suggested by stuart sir... and as ian has rightly mentioned, i also believe its a badly written question..!!!
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by lunarpower » Wed May 18, 2011 9:48 pm
received a PM.

i think the other experts on this thread have already sewed it up quite well, so there is little more to say.
however, this is a good opportunity for me to weigh in with my usual comment: beware obscure third-party or unsourced problems; these problems are, frankly, MUCH more likely to derail or even damage your studies than to help you.

almost *all* problems that i've seen from unsourced third-party materials have at least one of the following issues:
(a) they are actually incorrect;
(b) they test material that the official test doesn't test; or
(c) they test material with priorities or emphases that are fundamentally different from those of the official test.

so, beware.
if you've downloaded a bunch of questions from the internet for $0.00, then, in most cases, that is exactly how much they are actually worth.
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by pemdas » Wed May 18, 2011 10:11 pm
let me put it straight statistics way - out of the pool of 100 events each has a chance of 0.5 denotes the expected average, or E probability. It's not individual probability for each event, otherwise the problem stated simply 'Each lottery has winning chance 50%' and not 'Out of 100 lottery tickets, each having 50% chance'.

let me turn this question to English grammar context (btw, GMAT SC context) 'Three experts, each one posting in this thread, assume condition of the independent events.' <- each is part of plural subject, i.e. pool.

the pool changes, our expected average (E probability) must change.
champmag wrote:Out of 100 lottery tickets, each having 50% chance of winning, 2 tickets are bought at random. What is the probability that at least one ticket will win?

A.1/4
B.1/2
C.49/198
D.149/198
E.3/4
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by lunarpower » Wed May 18, 2011 10:24 pm
Pemdas, you are now arguing against *three* native English speakers who also happen to be experts on this material. Think about it -- when that happens, you are probably the one who's wrong.

Also, in earlier posts we've already noted that the wording of the question is unnecessarily ambiguous, and that you should ignore the question -- because the question is not going to contribute anything to your understanding of the test.
in fact -- as we are seeing right here, right now -- this question is currently serving to decrease your understanding, because you are becoming more and more convinced of a position that is actually invalid.

if you actually want to learn things for the test -- rather than defend an emotional investment in your argument -- then please ignore the wording of this problem, starting right now.
pemdas wrote:let me put it straight statistics way - out of the pool of 100 events each has a chance of 0.5 denotes the expected average, or E probability. It's not individual probability for each event, otherwise the problem stated simply 'Each lottery has winning chance 50%' and not 'Out of 100 lottery tickets, each having 50% chance'.
those two statements are indistinguishable, except in that the second also specifies the total number of tickets.
for your interpretation to be correct, the wording would have to say something like "...out of 100 lottery tickets, 50 of which are winners".

the current version says "each ticket has a 50% chance of winning" -- so that's what the current version means.
this is not the same as "exactly 50 tickets are winners", so that's the wrong interpretation.

this is the last post i will make on this topic, because there's nothing more to say about it -- but i can tell you with 100.000% confidence that:
* the position of the experts in this thread is correct;
* your current position is incorrect;
* this question is worthless, will cause nothing but confusion, and should be ignored.
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by pemdas » Thu May 19, 2011 12:04 am
thank you, i ignore this question and quit the argumentation what so ever
indeed we are hazed by many non-official stuff for GMAT
i appreciate other experts help too
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