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# When a random experiment is conducted, the probability that

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When a random experiment is conducted, the probability that [#permalink]  26 Sep 2010, 08:38
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When a random experiment is conducted, the probability that event A occurs is $$\frac{1}{3}$$. If the random experiment is conducted 5 independent times, what is the probability that event A occurs exactly twice?

A. 5/243
B. 25/243
C. 64/243
D. 80/243
E. 16/17

took more time than I hoped for in the tests (Kaplan). Don't forget to KUDOS if you like the question.

Tx.
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Re: Coin Toss [#permalink]  26 Sep 2010, 08:48
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Expert's post
hemanthp wrote:
When a random experiment is conducted, the probability that event A occurs is $$\frac{1}{3}$$. If the random experiment is conducted 5 independent times, what is the probability that event A occurs exactly twice?

5/243
25/243
64/243
80/243
16/17

took more time than I hoped for in the tests (Kaplan). Don't forget to KUDOS if you like the question.

Tx.

The probability that event A occurs is $$\frac{1}{3}$$;
The probability that event A will not occur is $$1-\frac{1}{3}=\frac{2}{3}$$.

We want to calculate the probability of event YYNNN (Y-occurs, N-does not occur).

$$P(YYNNN)=\frac{5!}{2!3!}*(\frac{1}{3})^2*(\frac{2}{3})^3=\frac{80}{243}$$, we are multiplying by $$\frac{5!}{2!3!}$$ as event YYNNN can happen in # of times: YYNNN, YNYNN, NNNYY, ... basically the # of permutations of 5 letters YYNNN out of which 2 Y's and 3 N's are identical ($$\frac{5!}{2!3!}$$).

Hope it's clear.
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Re: Coin Toss [#permalink]  26 Sep 2010, 08:56
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Expert's post
This can be broken down into two parts:

1. The ways in which two of these events can occur - Combination
2. The probability of the event occurring.

For part 1: We have 2 events out of 5 events and hence the ways you can choose two out of five events is: $$\frac{5!}{2! * 3!} = 10$$

For part 2: Probability of correct coin toss $$= \frac{1}{3}$$ and probability of not happening = $$1 - \frac{1}{3} = \frac{2}{3}$$

Hence required probability $$= \frac{1}{3} * \frac{1}{3} * \frac{2}{3}* \frac{2}{3}* \frac{2}{3} * 10 = \frac{80}{243}$$

Hope this helps.
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Re: Coin Toss [#permalink]  26 Sep 2010, 19:31
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1. number of ways to choose 2 occasion in which A would occur is 5c2=10
2. out of 5 times, the probability of occurring A is (1/3)(1/3)(1-2/3)(1-2/3)(1-2/3)=8/243

So the final probability is 10*(8/243)=80/243
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Re: Coin Toss [#permalink]  27 Sep 2010, 00:28
1
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Thank you guys! I felt this question was interesting.
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Re: Coin Toss [#permalink]  27 Sep 2010, 12:00
Awesome question, it tests important fundamentals
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Re: Coin Toss [#permalink]  28 Sep 2010, 02:18
Bunuel wrote:
hemanthp wrote:
When a random experiment is conducted, the probability that event A occurs is $$\frac{1}{3}$$. If the random experiment is conducted 5 independent times, what is the probability that event A occurs exactly twice?

5/243
25/243
64/243
80/243
16/17

took more time than I hoped for in the tests (Kaplan). Don't forget to KUDOS if you like the question.

Tx.

The probability that event A occurs is $$\frac{1}{3}$$;
The probability that event A will not occur is $$1-\frac{1}{3}=\frac{2}{3}$$.

We want to calculate the probability of event YYNNN (Y-occurs, N-does not occur).

$$P(YYNNN)=\frac{5!}{2!3!}*(\frac{1}{3})^2*(\frac{2}{3})^3=\frac{80}{243}$$, we are multiplying by $$\frac{5!}{2!3!}$$ as event YYNNN can happen in # of times: YYNNN, YNYNN, NNNYY, ... basically the # of permutations of 5 letters YYNNN out of which 2 Y's and 3 N's are identical ($$\frac{5!}{2!3!}$$).

Hope it's clear.

Great explanation.Thanks!
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Re: When a random experiment is conducted, the probability that [#permalink]  18 Jul 2015, 05:22
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Re: When a random experiment is conducted, the probability that [#permalink]  18 Jul 2015, 05:41
Expert's post
hemanthp wrote:
When a random experiment is conducted, the probability that event A occurs is $$\frac{1}{3}$$. If the random experiment is conducted 5 independent times, what is the probability that event A occurs exactly twice?

A. 5/243
B. 25/243
C. 64/243
D. 80/243
E. 16/17

took more time than I hoped for in the tests (Kaplan). Don't forget to KUDOS if you like the question.

Tx.

Probability of The event to occur = 1/3
i.e. Probability of The event to NOT occur = 1-(1/3) = (2/3)

Let Occurrence is represented by 'O'
and Non-Occurrence is represented by 'N'

Then the total ways of two occurrences out of 5 trials is same as arrangement of 5 letters 'OONNN'

Arrangement of 'OONNN' = 5C2 or 5!/(2!)(3!) = 10

i.e. The probability of two occurrences and 3 non-occurrences for one arrangement = (1/3)(1/3)(2/3)(2/3)(2/3) = (1/3)^2*(2/3)^3

i.e. The probability of two occurrences and 3 non-occurrences for all the arrangement = 5C2*(1/3)^2*(2/3)^3 = 10*(1/3)^2*(2/3)^3 = 80/243

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Re: When a random experiment is conducted, the probability that   [#permalink] 18 Jul 2015, 05:41
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