 # Assume the random variable X has a binomial distribution with Yasmin 2021-10-02 Answered

Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success . Round your answer to four decimal places. $$\displaystyle{P}{\left({X}{<}{4}\right)},{n}={6},{p}={0.4}$$

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Step 1
The Binomial probability distribution is:
In the formula, n denotes the number of trails, p denotes probability of success, and x denotes the number of success.
Step 2
The value of $$\displaystyle{P}{\left({X}{<}{4}\right)}$$ is,
$$\displaystyle{P}{\left({X}{<}{4}\right)}={P}{\left({X}={0}\right)}+{P}{\left({X}={1}\right)}+{P}{\left({X}={2}\right)}+{P}{\left({X}={3}\right)}$$
$$=\left(\begin{array}{c}6\\ 0\end{array}\right)(0.4)^{0}(1-0.4)^{6-0}+\left(\begin{array}{c}6\\ 0\end{array}\right)(0.4)^{1}(1-0.4)^{6-1}+\left(\begin{array}{c}6\\ 2\end{array}\right)(0.4)^{2}(1-0.4)^{6-2}+\left(\begin{array}{c}6\\ 3\end{array}\right)(0.4)^{3}(1-0.4)^{6-3}$$
$$\displaystyle={0.0467}+{0.1866}+{0.3110}+{0.2765}$$
$$\displaystyle={0.8208}$$
Thus, the value of $$\displaystyle{P}{\left({X}{<}{4}\right)}$$ is 0.8208.