Probability

We know that $n=4$ (there are 4 trials) and $P(X=0)=\;\frac{16}{81}$ (the chance of getting zero successes is $\;\frac{16}{81}$ ).In a binomial distribution, $p$ is the chance of success each time, and $q$ is the chance of failure.$q=1-p$.The formula to find the probability of getting $k$ successes is: $P(X=k)={}^n C_k p^k q^{n-k}$.Step 1: Find $q$$P(X=0)={}^4 C_0 p^0 q^4$${}^4 C_0=1$ and $p^0=1$, so:$P(X=0)=1 \cdot 1 \cdot q^4=q^4$We know $q^4=\frac{16}{81}$.To find $q$, take the fourth root of both sides:$q=\;\frac{2}{3}$ because $(\;\frac{2}{3})^4=\;\frac{16}{81}$.Step 2: Find $p$Since $q=\;\frac{2}{3}$,$p=1-\frac{2}{3}=\frac{1}{3}$.Step 3: Find $P(X=4)$$P(X=4)$ is the probability of getting 4 successes out of 4 tries.$P(X=4)={}^4 C_4 p^4 q^0$${}^4 C_4=1$ and $q^0=1$,So:$P(X=4)=1 \cdot \left(\;\frac{1}{3}\right)^4 \cdot 1= \left(\;\frac{1}{3}\right)^4$Calculate $\left(\;\frac{1}{3}\right)^4=\;\frac{1}{81}$.