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Math: Probability

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Re: Math: Probability  [#permalink]

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New post 12 Mar 2014, 11:22
Dear Walker

I was trying your given example- Given that there are 5 married couples. If we select only 3 people out of the 10, what is the probability that none of them are married to each other? with other method but the answer is coming wrong.

I did 10c1X8c1X6c1/ 10c3. We can select any one random person from 10 then to avoid pairing we remove his/her counterpart so next selection becomes 8c1 and similarly 6c1...but the probability is coming out to be 4. Where am I wrong?

Thanks for your help! :)
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Re: Math: Probability  [#permalink]

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New post 13 Mar 2014, 02:00
ShantnuMathuria wrote:
Dear Walker

I was trying your given example- Given that there are 5 married couples. If we select only 3 people out of the 10, what is the probability that none of them are married to each other? with other method but the answer is coming wrong.

I did 10c1X8c1X6c1/ 10c3. We can select any one random person from 10 then to avoid pairing we remove his/her counterpart so next selection becomes 8c1 and similarly 6c1...but the probability is coming out to be 4. Where am I wrong?

Thanks for your help! :)


The following post might help: if-4-people-are-selected-from-a-group-of-6-married-couples-99055.html#p764040

Also check similar questions to practice:
a-committee-of-three-people-is-to-be-chosen-from-four-teams-130617.html
if-4-people-are-selected-from-a-group-of-6-married-couples-99055.html
a-committee-of-3-people-is-to-be-chosen-from-four-married-94068.html
if-a-committee-of-3-people-is-to-be-selected-from-among-88772.html
a-comittee-of-three-people-is-to-be-chosen-from-four-married-130475.html
a-committee-of-three-people-is-to-be-chosen-from-4-married-101784.html
a-group-of-10-people-consists-of-3-married-couples-and-113785.html
if-there-are-four-distinct-pairs-of-brothers-and-sisters-99992.html
given-that-there-are-6-married-couples-if-we-select-only-58640.html
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Re: Math: Probability  [#permalink]

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New post 25 Feb 2017, 23:21
Example #2
Q: Given that there are 5 married couples. If we select only 3 people out of the 10, what is the probability that none of them are married to each other?
Solution:

1) combinatorial approach:
\(C^5_3\) - we choose 3 couples out of 5 couples.
\(C^2_1\) - we chose one person out of a couple.
\((C^2_1)^3\) - we have 3 couple and we choose one person out of each couple.
\(C^{10}_3\) - the total number of combinations to choose 3 people out of 10 people.

\(p=\frac{C^5_3*(C^2_1)^3}{C^{10}_3}=\frac{10*8}{10*3*4} = \frac{2}{3}\)

2) reversal combinatorial approach: In this example reversal approach is a bit shorter and faster.
\(C^5_1\) - we choose 1 couple out of 5 couples.
\(C^8_1\) - we chose one person out of remaining 8 people.
\(C^{10}_3\) - the total number of combinations to choose 3 people out of 10 people.

\(p=1 - \frac{C^5_1*C^8_1}{C^{10}_3}=1 - \frac{5*8}{10*3*4} = \frac{2}{3}\)

3) probability approach:
1st person: \(\frac{10}{10} = 1\) - we choose any person out of 10.
2nd person: \(\frac{8}{9}\) - we choose any person out of 8=10-2(one couple from previous choice)
3rd person: \(\frac{6}{8}\) - we choose any person out of 6=10-4(two couples from previous choices).

\(p = 1*\frac{8}{9}*\frac{6}{8}=\frac{2}{3}\)


--------------------------------------------------------


Didn't follow the reverse combinatorial approach here. The question asks the probability that none of the 3 people selected would be married. But the approach is taking 1 married couple and 1 unmarried person. What am I missing?
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Re: Math: Probability  [#permalink]

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New post 05 Jul 2017, 15:26
Can any one help me solve below example and how the probability tree works?

Q: Julia and Brian play a game in which Julia takes a ball and if it is green, she wins. If the first ball is not green, she takes the second ball (without replacing first) and she wins if the two balls are white or if the first ball is gray and the second ball is white. What is the probability of Julia winning if the jar contains 1 gray, 2 white and 4 green balls?
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Re: Math: Probability  [#permalink]

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New post 19 Jul 2017, 20:19
This is very basic but I am getting confused by the concepts.

If a fair coin is flipped three times, what is the probability that it
comes up heads all three times?

So the way I understand it, P of getting tails is 1/2*1/2*1/2=1/8

Because each event is independent from one another, I calculate the probability of getting heads as 1-1/8=7/8 but the right answer is 1/8....

What am I doing wrong?
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Re: Math: Probability  [#permalink]

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New post 22 Jul 2017, 12:51
ganand wrote:
azelastine wrote:
This is very basic but I am getting confused by the concepts.

If a fair coin is flipped three times, what is the probability that it
comes up heads all three times?

So the way I understand it, P of getting tails is 1/2*1/2*1/2=1/8

Because each event is independent from one another, I calculate the probability of getting heads as 1-1/8=7/8 but the right answer is 1/8....

What am I doing wrong?


Hi azelastine,

Your first part is correct, i.e. probability of getting 3 tails is 1/8.
Similarly, you can compute the probability of getting three heads.
Please note that each coin toss is independent of one another, that's why we can multiply the probability of each event.
P(HHH) = P(getting head on the first toss) and P(getting head on the second toss) and P(getting head on the third toss)= 1/2 * 1/2 * 1/2 = 1/8

AND implies Multiplication(*)
OR implies Addition(+)

Quote:
Because each event is independent from one another, I calculate the probability of getting heads as 1-1/8=7/8 but the right answer is 1/8....


Here you are assuming that following events are complementary to each other:
1. Getting three tails, i.e. P(TTT) and
2. Getting three heads, i.e.P(HHH)
P(TTT) + P(HHH) = 1
This is not correct.

Let's enumerate the all possible outcomes(sample space).
1. HHH
2. HHT
3. HTT
4. TTT
5. TTH
6. THH
7. THT
8. HTH

Sum of probabilities of all these events will be equal to 1. i.e.
P(HHH) + P(HHT) + P(HTT) + P(TTT) + P(TTH) + P(THH) + P(THT) + P(HTH) = 1

In general, if you toss the coin n times, then total possible outcomes = 2^n. In this cane, n= 3, so total outcomes = 2^3 = 8.

Hope it helps.

Thanks.


Thanks for the explanation, this is very helpful.

Could you help me understand the difference between this and the following problem?

A fair coin is flipped twice. What is the probability that the coin lands showing heads on the first flip, second flip or both?

The solution to this question is two-step: 1) P (two tails)=1/4, 2) 1-1/4=3/4
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Re: Math: Probability  [#permalink]

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New post 28 Jul 2017, 04:30
azelastine wrote:
Thanks for the explanation, this is very helpful.

Could you help me understand the difference between this and the following problem?

A fair coin is flipped twice. What is the probability that the coin lands showing heads on the first flip, second flip or both?

The solution to this question is two-step: 1) P (two tails)=1/4, 2) 1-1/4=3/4


Hi azelastine,

I'll try to explain the difference.

Quote:
A fair coin is flipped twice. What is the probability that the coin lands showing heads on the first flip, second flip or both?


Sample space: {HH, HT, TH, TT}

Favorable event = {HH, HT, TH} and complementary event = {TT}

In two ways you can solve this question.

1. Compute the probability of the favorable event
In this case required probability = P(HH) + P(HT) + P(TH) =\(\frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} = \frac{3}{4}\)
or
2. Compute the probability of complementary event and subtract it from 1. You have solved it using this method.

In general, when a complementary event is small then use the 2nd approach.
The complementary approach can also be used in P&C questions.

Please go through the first post of the thread. These points are discussed in detail with examples.

Hope this helps.
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Re: Math: Probability  [#permalink]

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New post 12 Sep 2017, 02:06
Q: If the probability of a certain event is p, what is the probability of it occurring k times in n-time sequence?
(Or in English, what is the probability of getting 3 heads while tossing a coin 8 times?)
Solution: All events are independent. So, we can say that:

P′=pk∗(1−p)n−kP′=pk∗(1−p)n−k (1)

But it isn't the right answer. It would be right if we specified exactly each position for events in the sequence. So, we need to take into account that there are more than one outcomes. Let's consider our example with a coin where "H" stands for Heads and "T" stands for Tails:
HHHTTTTT and HHTTTTTH are different mutually exclusive outcomes but they both have 3 heads and 5 tails. Therefore, we need to include all combinations of heads and tails. In our general question, probability of occurring event k times in n-time sequence could be expressed as:

P=Cnk∗pk∗(1−p)n−kP=Ckn∗pk∗(1−p)n−k (2)

In the example with a coin, right answer is P=C83∗0.53∗0.55=C83∗0.58

Aren't we looking for the permutations here instead of the combination ? aren't HHHTTTTT and HHTTTTTH the same combination but two different permutations?
Thank you for your clarifications ;)
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Math: Probability  [#permalink]

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New post 18 Jul 2018, 20:17
walker wrote:


2) reversal combinatorial approach: Instead of counting probability of occurrence of certain event, sometimes it is better to calculate the probability of the opposite and then use formula p = 1 - q. The total number of possible committees is \(N=C^8_2\). The number of possible committee that does not includes both Bob and Rachel is:
\(m = C^6_2 + 2*C^6_1\) where,
\(C^6_2\) - the number of committees formed from 6 other people.
\(2*C^6_1\) - the number of committees formed from Rob or Rachel and one out of 6 other people.



Could you please confirm if 2 in 2*C^6_1 above is the result of C^2_1?
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Math: Probability  [#permalink]

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New post 07 Jan 2019, 21:34
Bunuel wrote:
ShantnuMathuria wrote:
Dear Walker

I was trying your given example- Given that there are 5 married couples. If we select only 3 people out of the 10, what is the probability that none of them are married to each other? with other method but the answer is coming wrong.

I did 10c1X8c1X6c1/ 10c3. We can select any one random person from 10 then to avoid pairing we remove his/her counterpart so next selection becomes 8c1 and similarly 6c1...but the probability is coming out to be 4. Where am I wrong?

Thanks for your help! :)


The following post might help: http://gmatclub.com/forum/if-4-people-a ... ml#p764040

Also check similar questions to practice:
http://gmatclub.com/forum/a-committee-o ... 30617.html
http://gmatclub.com/forum/if-4-people-a ... 99055.html
http://gmatclub.com/forum/a-committee-o ... 94068.html
http://gmatclub.com/forum/if-a-committe ... 88772.html
http://gmatclub.com/forum/a-comittee-of ... 30475.html
http://gmatclub.com/forum/a-committee-o ... 01784.html
http://gmatclub.com/forum/a-group-of-10 ... 13785.html
http://gmatclub.com/forum/if-there-are- ... 99992.html
http://gmatclub.com/forum/given-that-th ... 58640.html



Total outcomes= 10C3=120

Favourable outcomes= (5C2x3C1 + 5C2x3C1 + 5C3 +5C3) = (30 + 30 + 10 + 10) = 80
[Since, if we select 2 females from 5 couple pairs then we have to choose 1 male from remaining 3 pairs=>none are married to each other ; In the same way we can choose 2 males and 1 female ; we can also choose 3 males out of 5 pairs=>none are married to each other ; again we can choose 3 females out of 5 pairs ; all the events are mutually exclusive so we add them;]

P(selected 3 ppl are not married to each other)=80/120=2/3

Bunuel Is my way of approaching this problem right??
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Re: Math: Probability  [#permalink]

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New post 20 Jul 2019, 11:54
I am having trouble understanding the combinatorial and reversal combinatorial approach in terms of the logic of calculating the notation from this explanation.

Example #1 under "A few ways to approach a probability problem": \(P=\frac{n}{N}\) where N = \(C |_2^8\) I am not understanding at all how you go from \(C |_2^8\) to 28. Would anyone be willing to help explain what I am missing? At first I thought it was factorials \(\frac{8!}{2!}\) but that does not equal 28. I understand the regular probability approach and reversal probability approach, but right now am completely lost on the combinatorial.

Any help explaining would be greatly appreciated!
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Re: Math: Probability   [#permalink] 20 Jul 2019, 11:54

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