Question

President Biden’s approval rating is 42%. Suppose that 8 people were chosen at randoma) Find the probability that 5 of the 8 people approve of the job President Biden is doing.b) Find the probability that at most 3 of the 8 people approve of the job President Biden is doing.c) Find the probability that at least 3 of 8 people approve of the job President Biden is doing.

Answer 1

Given:

Approval rating = 42%

Number of people chosen = 8

Let's solve for the following:

• (a). Find the probability that 5 of the 8 people approve of the job President Biden is doing.

Here, we are to apply Binomial distribution.

Apply the formula:

$P\left(X=x\right)=^nC_x*p^x*1-p^{n-x}$

We are to solve for P(X = 5).

Where:

n = 8

x = 5

p = 42% = 0.42

We have:

$\begin{gathered} P(X=5)=^8C_5*0.42^5*(1-0.42^)^{8-5} \\ \\ \end{gathered}$

Solving further:

$\begin{gathered} P(X=5)=(\frac{8!}{5!(8-5)!})*0.013069*(0.58)^3 \\ \\ P(X=5)=56*0.013069*0.195112 \\ \\ P(X=5)=0.143 \end{gathered}$

Therefore, the probability that 5 of the 8 people approve is 0.143

• (b). , Find the probability that at most 3 of the 8 people approve of the job President Biden is doing.

Here we are to solve for P(X≤ 3).

We have:

$\begin{gathered} P(X\leq3)=(^8C_0*0.42^0*0.58^8)+(^8C_1*0.42^1*0.58^7)+(^8C_2*0.42^2*0.58^6)+(^8C_3*0.42^3*0.58^5) \\ \\ P(X\leq3)=0.0128+0.0742+0.1880+0.2723 \\ \\ P(X\leq3)=0.5473 \end{gathered}$

Therefore, the probability that at most 3 people approve is 0.5473

• (c)., Find the probability that at least 3 of 8 people approve of the job President Biden is doing.

We have:

$P(x\ge3)=(^8C_3*0.42^3*0.58^5)+(^8C_4*0.42^4*0.58^4)+(^8C_5*0.42^5*0.58^3)+(^8C_6*0.42^6*0.58^2)+(^8C_7*0.42^7*0.58^1)+(^8C_8*0.42^8*0.58^0)$

Solving further:

P(x ≥ 3) = 0.2723 + 0.2465 + 0.1428 + 0.0517 + 0.0107 + 0.000968 = 0.7250

ANSWER:

• (a). 0.143

• (b). 0.5473

• (c). 0.7250