Question

suppose a random variable, x, arises from a binomial experiment. suppose n = 6, and p = 0.12. write the probability distribution. round to six decimal places, if necessary.

x

p(x)

|0|
|1|
|2|
|3|
|4|
|5|
|6|
select the correct histogram.

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $p$ is the probability of success on a single - trial, $k$ is the number of successes, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 6$ and $p=0.12$, so $1 - p = 0.88$.

Step2: Calculate $P(X = 0)$

$C(6,0)=\frac{6!}{0!(6 - 0)!}=1$. Then $P(X = 0)=C(6,0)\times(0.12)^{0}\times(0.88)^{6}=1\times1\times0.88^{6}\approx0.464404$.

Step3: Calculate $P(X = 1)$

$C(6,1)=\frac{6!}{1!(6 - 1)!}=\frac{6!}{1!5!}=6$. Then $P(X = 1)=C(6,1)\times(0.12)^{1}\times(0.88)^{5}=6\times0.12\times0.88^{5}\approx0.379399$.

Step4: Calculate $P(X = 2)$

$C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6\times5}{2\times1}=15$. Then $P(X = 2)=C(6,2)\times(0.12)^{2}\times(0.88)^{4}=15\times0.0144\times0.88^{4}\approx0.139317$.

Step5: Calculate $P(X = 3)$

$C(6,3)=\frac{6!}{3!(6 - 3)!}=\frac{6\times5\times4}{3\times2\times1}=20$. Then $P(X = 3)=C(6,3)\times(0.12)^{3}\times(0.88)^{3}=20\times0.001728\times0.88^{3}\approx0.025439$.

Step6: Calculate $P(X = 4)$

$C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. Then $P(X = 4)=C(6,4)\times(0.12)^{4}\times(0.88)^{2}=15\times0.00020736\times0.88^{2}\approx0.002730$.

Step7: Calculate $P(X = 5)$

$C(6,5)=\frac{6!}{5!(6 - 5)!}=6$. Then $P(X = 5)=C(6,5)\times(0.12)^{5}\times(0.88)^{1}=6\times2.48832\times10^{- 5}\times0.88\approx0.000131$.

Step8: Calculate $P(X = 6)$

$C(6,6)=\frac{6!}{6!(6 - 6)!}=1$. Then $P(X = 6)=C(6,6)\times(0.12)^{6}\times(0.88)^{0}=1\times2.985984\times10^{-6}\times1\approx0.000003$.

Answer:

x	P(x)
1	0.379399
2	0.139317
3	0.025439
4	0.002730
5	0.000131
6	0.000003