If in a binomial distribution n = 4 and P(X = 0) =16 81 , find q. - Vidyadip

Question

If in a binomial distribution n = 4 and P(X = 0) $=\frac{16}{81},$ find q.

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Answer

Given that,
$\text{n}=4,\text{P(X}=0)=\frac{16}{81}$
We know that,
$\text{P(X = r})=\text{ }^{\text{n}}\text{c}_{\text{r}}\text{p}^{\text{r}}\text{q}^{\text{n}-\text{r}}$
$\text{P(X = r})=\text{ }^4\text{c}_{\text{r}}(\text{p})^{\text{r}}(\text{q})^{4-\text{r}}$
$\text{P(X}=0)=\text{ }^4\text{c}_0(1-\text{q})^0(\text{q})^{4-0}$
$\frac{16}{81}=1.1.\text{q}^4$
$\text{q}^4=\big(\frac{2}{3}\big)^4$
$\text{q}=\frac{2}{3}$

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