Question

question 4 suppose a random variable, x, arises from a binomial experiment. if n = 10, and p = 0.2, find the following probabilities using the binomial formula. round to four decimal places, if necessary. p(x = 4) = p(x = 9) = p(x = 3) = p(x ≤ 7) = p(x ≥ 5) = p(x ≤ 8) =

Explanation:

Step1: Recall binomial formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $C(n,k)=\frac{n!}{k!(n - k)!}$, $n$ is the number of trials, $p$ is the probability of success on a single - trial, and $k$ is the number of successes. Also, $P(X\leq m)=\sum_{k = 0}^{m}C(n,k)\times p^{k}\times(1 - p)^{n - k}$ and $P(X\geq m)=1 - P(X\leq m - 1)$.

Step2: Calculate $P(x = 4)$

$C(10,4)=\frac{10!}{4!(10 - 4)!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210$.
$P(x = 4)=C(10,4)\times(0.2)^{4}\times(0.8)^{6}=210\times0.0016\times0.262144\approx0.0881$.

Step3: Calculate $P(x = 9)$

$C(10,9)=\frac{10!}{9!(10 - 9)!}=10$.
$P(x = 9)=C(10,9)\times(0.2)^{9}\times(0.8)^{1}=10\times5.12\times10^{-7}\times0.8\approx4.10\times10^{-6}$.

Step4: Calculate $P(x = 3)$

$C(10,3)=\frac{10!}{3!(10 - 3)!}=\frac{10\times9\times8}{3\times2\times1}=120$.
$P(x = 3)=C(10,3)\times(0.2)^{3}\times(0.8)^{7}=120\times0.008\times0.2097152\approx0.2013$.

Step5: Calculate $P(x\leq7)$

$P(x\leq7)=1 - P(x = 8)-P(x = 9)-P(x = 10)$.
$C(10,8)=\frac{10!}{8!(10 - 8)!}=45$, $P(x = 8)=C(10,8)\times(0.2)^{8}\times(0.8)^{2}=45\times2.56\times10^{-6}\times0.64\approx7.37\times10^{-5}$.
$C(10,9)=10$, $P(x = 9)=10\times5.12\times10^{-7}\times0.8\approx4.10\times10^{-6}$.
$C(10,10)=1$, $P(x = 10)=(0.2)^{10}=1.024\times10^{-7}$.
$P(x\leq7)=1-(7.37\times10^{-5}+4.10\times10^{-6}+1.024\times10^{-7})\approx0.9992$.

Step6: Calculate $P(x\geq5)$

$P(x\geq5)=1 - P(x\leq4)$.
$P(x\leq4)=P(x = 0)+P(x = 1)+P(x = 2)+P(x = 3)+P(x = 4)$.
$C(10,0)=1$, $P(x = 0)=(0.8)^{10}=0.1074$.
$C(10,1)=\frac{10!}{1!(10 - 1)!}=10$, $P(x = 1)=10\times0.2\times(0.8)^{9}=10\times0.2\times0.134217728 = 0.2684$.
$C(10,2)=\frac{10!}{2!(10 - 2)!}=45$, $P(x = 2)=45\times(0.2)^{2}\times(0.8)^{8}=45\times0.04\times0.16777216=0.3020$.
$P(x\leq4)=0.1074 + 0.2684+0.3020 + 0.2013+0.0881=0.9672$.
$P(x\geq5)=1 - 0.9672 = 0.0328$.

Step7: Calculate $P(x\leq8)$

$P(x\leq8)=1 - P(x = 9)-P(x = 10)=1-(4.10\times10^{-6}+1.024\times10^{-7})\approx0.999996$.

Answer:

$P(x = 4)\approx0.0881$
$P(x = 9)\approx4.10\times10^{-6}$
$P(x = 3)\approx0.2013$
$P(x\leq7)\approx0.9992$
$P(x\geq5)\approx0.0328$
$P(x\leq8)\approx0.999996$