question 4
in 5 tosses of a fair coin, what is the probability of exactly 3 heads?
0.313
0.078
0.500
0.188

Question

Accepted Answer

# Explanation:
## Step1: Determine the 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, $k$ is the number of successful trials, $p$ is the probability of success in a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$

## Step2: Identify the values of $n$, $k$, and $p$
Here, $n = 5$ (number of coin - tosses), $k=3$ (number of heads), and $p = 0.5$ (probability of getting a head in a single fair - coin toss)

## Step3: Calculate the combination $C(n,k)$
$$
\begin{align*}
C(5,3)&=\frac{5!}{3!(5 - 3)!}\
&=\frac{5!}{3!2!}\
&=\frac{5\times4\times3!}{3!\times2\times1}\
&=10
\end{align*}
$$

## Step4: Calculate $p^{k}$ and $(1 - p)^{n - k}$
$p^{k}=(0.5)^{3}=\frac{1}{8}$, $(1 - p)^{n - k}=(0.5)^{2}=\frac{1}{4}$

## Step5: Calculate the probability $P(X = 3)$
$$
\begin{align*}
P(X = 3)&=C(5,3)\times p^{3}\times(1 - p)^{2}\
&=10\times\frac{1}{8}\times\frac{1}{4}\
&=\frac{10}{32}\
& = 0.3125\approx0.313
\end{align*}
$$

# Answer:
0.313

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QUESTION IMAGE

Question

Explanation:

Step1: Determine the binomial probability formula

Step2: Identify the values of \(n\), \(k\), and \(p\)

Step3: Calculate the combination \(C(n,k)\)

Step4: Calculate \(p^{k}\) and \((1 - p)^{n - k}\)

Step5: Calculate the probability \(P(X = 3)\)

Answer: