Using the formula for the multinomial distribution, the probability of $\mathrm { X } 1 = 2 , \mathrm { X } 2 = 2 , \mathrm { X } 3 = 1$ when $\mathrm { n } = 5$, $\mathrm { p } 1 = 0.4 , \mathrm { p } 2 = 0.5 , \mathrm { p } 3 = 0.1$ is
A) $0.0600$
B) $0.0900$
C) $0.1200$
D) $0.1800$

Question

Quizplus · Accepted Answer

The formula for the multinomial distribution is:

$$P\left(X_{1}=x_{1}, X_{2}=x_{2}, \ldots, X_{k}=x_{k}\right)=\frac{n !}{x_{1} ! x_{2} ! \cdots x_{k} !} p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{k}^{x_{k}}$$

where:
\begin{itemize}
\item $n$ is the total number of trials or experiments.
\item $X_i$ is the number of times outcome $i$ occurs.
\item $x_i$ is a specific value of $X_i$ that we want to find the probability of.
\item $p_i$ is the probability of outcome $i$ on any given trial.
\end{itemize}

Here, $n=5$, $X_1=2$, $X_2=2$, $X_3=1$, $p_1=0.4$, $p_2=0.5$, and $p_3=0.1$.

So, the probability is:

\begin{align*}
&P\left(X_{1}=2, X_{2}=2, X_{3}=1\right)\
&=\frac{5!}{2!2!1!}(0.4)^2(0.5)^2(0.1)^1 \
&= 0.12
\end{align*}

Therefore, the answer is C.

Using the Formula for the Multinomial Distribution, the Probability Of \mathrm