# An experiment succeeds twice as often as it fails.

Question:

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:

The probability of success is twice the probability of failure.

Let the probability of failure be x.

∴ Probability of success = 2x

$x+2 x=1$

$\Rightarrow 3 x=1$

$\Rightarrow x=\frac{1}{3}$

$\therefore 2 x=\frac{2}{3}$

Let $p=\frac{1}{3}$ and $q=\frac{2}{3}$

Let X be the random variable that represents the number of successes in six trials.

By binomial distribution, we obtain

$\mathrm{P}(\mathrm{X}=x)={ }^{n} \mathrm{C}_{x} p^{n-x} q^{x}$

Probability of at least 4 successes = P (X ≥ 4)

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

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

$=\frac{(2)^{4}}{(3)^{6}}[15+12+4]$

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

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