if a coin is flipped four times, find the probability distribution for the number of heads. listed below are…
if a coin is flipped four times, find the probability distribution for the number of heads. listed below are four charts. choose the chart that best evaluates the data. chart a\nx 0 1 2 3 4\np(x) $\frac{1}{16}$ $\frac{1}{8}$ $\frac{3}{8}$ $\frac{1}{8}$ $\frac{1}{16}$\nchart b\nx 1 2 3 4\np(x) $\frac{1}{4}$ $\frac{7}{16}$ $\frac{1}{4}$ $\frac{1}{16}$\nchart c\nx 0 1 2 3 4\np(x) $\frac{1}{16}$ $\frac{1}{4}$ $\frac{3}{8}$ $\frac{1}{4}$ $\frac{1}{16}$
Answer
Explanation:
Step1: Calculate total number of outcomes
A coin - flip has 2 possible outcomes. When flipped 4 times, the total number of outcomes is $2^4=16$ by the multiplication principle.
Step2: Use binomial probability formula
The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n = 4$ (number of trials), $p=\frac{1}{2}$ (probability of getting a head in a single - flip), and $C(n,k)=\frac{n!}{k!(n - k)!}$.
For $k = 0$:
$C(4,0)=\frac{4!}{0!(4 - 0)!}=1$, $P(X = 0)=C(4,0)\times(\frac{1}{2})^{0}\times(\frac{1}{2})^{4}=\frac{1}{16}$.
For $k = 1$:
$C(4,1)=\frac{4!}{1!(4 - 1)!}=\frac{4!}{1!3!}=4$, $P(X = 1)=C(4,1)\times(\frac{1}{2})^{1}\times(\frac{1}{2})^{3}=\frac{4}{16}=\frac{1}{4}$.
For $k = 2$:
$C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3\times2!}{2!\times2!}=6$, $P(X = 2)=C(4,2)\times(\frac{1}{2})^{2}\times(\frac{1}{2})^{2}=\frac{6}{16}=\frac{3}{8}$.
For $k = 3$:
$C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=4$, $P(X = 3)=C(4,3)\times(\frac{1}{2})^{3}\times(\frac{1}{2})^{1}=\frac{4}{16}=\frac{1}{4}$.
For $k = 4$:
$C(4,4)=\frac{4!}{4!(4 - 4)!}=1$, $P(X = 4)=C(4,4)\times(\frac{1}{2})^{4}\times(\frac{1}{2})^{0}=\frac{1}{16}$.
The probability distribution is:
| $x$ | $P(x)$ |
|---|---|
| 0 | $\frac{1}{16}$ |
| 1 | $\frac{1}{4}$ |
| 2 | $\frac{3}{8}$ |
| 3 | $\frac{1}{4}$ |
| 4 | $\frac{1}{16}$ |
So the correct chart is Chart C.