pareekbharat86 wrote:When a random experiment is conducted, the probability that event A occurs is 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. 16/27
P(exactly n times) = P(one way) * total possible ways.
Let A = a result of A and N = a result of not A.
Since P(A) = 1/3, P(N) = 2/3.
P(one way):
Over the course of 5 experiments, ONE WAY to get exactly 2 A's is AANNN.
P(A is yielded by the 1st experiment) = 1/3.
P(A is yielded by the 2nd experiment) = 1/3.
P(N is yielded by the 3rd experiment) = 2/3.
P(N is yielded by the 4th experiment) = 2/3.
P(N is yielded by the 5th experiment) = 2/3.
Since we want all of these events to happen, we MULTIPLY:
1/3 * 1/3 * 2/3 * 2/3 * 2/3 = 8/243.
Total possible ways:
AANNN is only ONE WAY to get exactly 2 A's.
Now we must account for ALL OF THE WAYS to get exactly 2 A's.
Any arrangement of the letters AANNN represents one way to get exactly 2 A's and 3 N's.
Thus, to account for ALL OF THE WAYS to get exactly 2 A's and 3 N's, the result above must be multiplied by the number of ways to arrange the letters AANNN.
Number of ways to arrange 5 elements = 5!.
But when an arrangement includes IDENTICAL elements, we must divide by the number of ways each set of identical elements can be ARRANGED.
The reason:
When the identical elements swap positions, the arrangement doesn't change.
Here, we must divide by 2! to account for the two identical A's and by 3! to account for the two identical N's:
5!/(2!3!) = 10.
Multiplying the results above, we get:
P(exactly 2 A's) = 10 * 8/243 = 80/243.
The correct answer is
D.
More practice:
https://www.beatthegmat.com/select-exact ... 88786.html
https://www.beatthegmat.com/probability- ... 14250.html
https://www.beatthegmat.com/a-single-par ... 28342.html
https://www.beatthegmat.com/at-a-blind-t ... 20058.html
https://www.beatthegmat.com/rain-check-t79099.html
https://www.beatthegmat.com/probability-t227448.htmlPrivate tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at
[email protected].
Student Review #1
Student Review #2
Student Review #3
