find the probability of having 2,3, or 4 successes in 5 trials of a binomial experiment in which the probability of success is 40%. Round to the nearest tenth of a percent.

Question

holsteremission · Answer

The probability of getting exactly $k$ successes out of $n$ trials with success probability $p$ is given by $\mathbb{P}(X=k)=\dbinom nk p^k(1-p)^{n-k}$.

So the probability of getting $2$, $3$, or $4$ successes (note that these are disjoint events) out of $5$ trials and with probability of success $p=\dfrac{2}{5}=40\%$ would be the summation of the probabilities of all of these events, or
$$\mathbb{P}(X=2~|~X=3~|~X=4)=\sum_{k=2}^4\binom 5k\left(\frac{2}{5}\right)^k\left(\frac{3}{5}\right)^{5-k}$$