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Home » An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.
An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.
CBSE | Class 12 | Maths | NCERT | Probability
An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.

Solution:

Given that probability of failure =x

And, probability of success =2 x

\therefore x+2 x=1

3 x=1

x=1 / 3

2 x=2 / 3

Let p=1 / 3 and q=2 / 3

In addition, let X be the random variable that represents the number of trials in the experiment.

As a result of the binomial distribution, we have:

P(X=x)={ }^{n} C_{x} p^{n-x} q^{x}

\therefore Probability of having at least 4 successes =P(X \geq 4)

=\mathrm{P}(\mathrm{X}=4)+\mathrm{P}(\mathrm{X}=5)+\mathrm{P}(\mathrm{X}=6)

={ }^{6} C_{4}\left(\frac{2}{3}\right)^{4}\left(\frac{1}{3}\right)^{2}+{ }^{6} C_{5}\left(\frac{2}{3}\right)^{5}\left(\frac{1}{3}\right)+{ }^{6} C_{6}\left(\frac{2}{3}\right)^{6}

=\frac{15(2)^{4}}{3^{6}}+\frac{6(2)^{5}}{3^{6}}+\frac{(2)^{6}}{3^{6}}

=\frac{31 \times(2)^{4}}{(3)^{6}}

=\frac{31}{9}\left(\frac{2}{3}\right)^{4}

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