Question

suppose a random variable, x, arises from a binomial experiment. suppose n = 7, and p = 0.45. write the probability distribution. round to six decimal places, if necessary.

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, $n = 7$ and $p=0.45$, so $1 - p = 0.55$.

Step2: Calculate probabilities for $k = 0$

$C(7,0)=\frac{7!}{0!(7 - 0)!}=1$. Then $P(X = 0)=C(7,0)\times(0.45)^{0}\times(0.55)^{7}=1\times1\times0.01525219\approx0.015252$.

Step3: Calculate probabilities for $k = 1$

$C(7,1)=\frac{7!}{1!(7 - 1)!}=\frac{7!}{1!6!}=7$. Then $P(X = 1)=C(7,1)\times(0.45)^{1}\times(0.55)^{6}=7\times0.45\times0.02773126\approx0.087203$.

Step4: Calculate probabilities for $k = 2$

$C(7,2)=\frac{7!}{2!(7 - 2)!}=\frac{7\times6}{2\times1}=21$. Then $P(X = 2)=C(7,2)\times(0.45)^{2}\times(0.55)^{5}=21\times0.2025\times0.05040809\approx0.214022$.

Step5: Calculate probabilities for $k = 3$

$C(7,3)=\frac{7!}{3!(7 - 3)!}=\frac{7\times6\times5}{3\times2\times1}=35$. Then $P(X = 3)=C(7,3)\times(0.45)^{3}\times(0.55)^{4}=35\times0.091125\times0.09150625\approx0.291734$.

Step6: Calculate probabilities for $k = 4$

$C(7,4)=\frac{7!}{4!(7 - 4)!}=35$. Then $P(X = 4)=C(7,4)\times(0.45)^{4}\times(0.55)^{3}=35\times0.04100625\times0.166375\approx0.242719$.

Step7: Calculate probabilities for $k = 5$

$C(7,5)=\frac{7!}{5!(7 - 5)!}=21$. Then $P(X = 5)=C(7,5)\times(0.45)^{5}\times(0.55)^{2}=21\times0.01845281\times0.3025\approx0.117271$.

Step8: Calculate probabilities for $k = 6$

$C(7,6)=\frac{7!}{6!(7 - 6)!}=7$. Then $P(X = 6)=C(7,6)\times(0.45)^{6}\times(0.55)^{1}=7\times0.00830376\times0.55\approx0.032711$.

Step9: Calculate probabilities for $k = 7$

$C(7,7)=\frac{7!}{7!(7 - 7)!}=1$. Then $P(X = 7)=C(7,7)\times(0.45)^{7}\times(0.55)^{0}=1\times0.00373669\times1\approx0.003737$.

Answer:

0: 0.015252
1: 0.087203
2: 0.214022
3: 0.291734
4: 0.242719
5: 0.117271
6: 0.032711
7: 0.003737