Probability
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Q.

An experiment succeeds twice as often as it fails. What is the probability that in the next 5 trials there will be four successes?

Note: As Sriram Gopal Goli points out in the comment, the experiment has only two possible outcomes, i.e. success or failure.

 A.

$0$

 B.

$\left(\frac{2}{3}\right)^4$

 C.

$5\times \left(\frac{2}{3}\right)^4\times \left(\frac{1}{3}\right)$

 D.

$ \left(\frac{2}{3}\right)^4\times \left(\frac{1}{3}\right)$

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Solution:
Option(C) is correct

An experiment succeeds twice as often as it fails. 

i.e. the probability of its success is \(\dfrac{2}{3}\) and the probability of its failure is \(\dfrac{1}{3}\)

In the next 5 trials the experiment needs to succeed in 4 out of the 5 trials. 

4 out of the 5 trials in which it succeeds could be selected in \(^5C_4=5\) ways.

And as 4 of them are successes, they have a probability of 2/3 and the one that is a failure will have a probability of

\(\dfrac{1}{3}\)

Hence, the required probability

\(=5\times \left(\dfrac{2}{3}\right)^4\times \left(\dfrac{1}{3}\right)\)


(2) Comment(s)


Sriram Gopal Goli
 ()

This problem is not complete unless it specifies the case where the experiment neither succeeds nor fails. Based on the solution, the probability for this is 0. But the problem must specify this.


Deepak
 ()

Thank you for letting me know, corrected the question.