Question

An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least $3$ successes

Answer 1

An experiment succeeds as often as it fails.

Therefore, there are 3 successes and 1 failure.

Thus the probability of success = $\frac{1}{3}$

And the probability of failure = $\frac{1}{4}$

We need to find the probability of atleast 3 successes in the next 5 trials.

Required probability = $P(X=3)+P(X=4)+P(X=5)$

$⟹{=}^5{C}_3{p}^3{q}^2{+}^5{C}_4{p}^4{q}^1{+}^5{C}_5{p}^5{q}^0⟹{=}^5{C}_3{\left(\frac{3}{4}\right)}^3{\left(\frac{1}{4}\right)}^2{+}^5{C}_4{\left(\frac{3}{4}\right)}^4{\left(\frac{1}{4}\right)}^1{+}^5{C}_5{\left(\frac{3}{4}\right)}^5{\left(\frac{1}{4}\right)}^0⟹=10{\left(\frac{3}{4}\right)}^3{\left(\frac{1}{4}\right)}^2+5{\left(\frac{3}{4}\right)}^3{\left(\frac{1}{4}\right)}^2+{\left(\frac{3}{4}\right)}^3{\left(\frac{1}{4}\right)}^2⟹\frac{918}{1024}=\frac{459}{512}=0.896$

is the required probability.