Answered

On a popular app, users rate hair salons as [tex]$1, 2, 3, 4,$[/tex] or [tex]$5$[/tex] stars. Suppose a rating is randomly selected from all the ratings on the app. Let [tex]$X$[/tex] be the number of stars of the selected rating. Here is the probability distribution of [tex]$X$[/tex].

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Value $x$ of $X$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$P ( X = x )$ & 0.24 & 0.20 & 0.09 & 0.20 & 0.27 \\
\hline
\end{tabular}
\][/tex]

For parts (a) and (b) below, find the probability that the randomly selected hair salon rating has the described number of stars.

(a) Fewer than 3: [tex]$\square$[/tex]

(b) 4 or more: [tex]$\square$[/tex]



Answer :

GinnyAnswer
Let's solve the problem step-by-step.

### Part (a): Probability of fewer than 3 stars

We need to find the probability that the rating is fewer than 3 stars. This means we are focusing on the probabilities of ratings 1 star and 2 stars.

The given probability distribution is:
- [tex]\( P(X = 1) = 0.24 \)[/tex]
- [tex]\( P(X = 2) = 0.20 \)[/tex]
- [tex]\( P(X = 3) = 0.09 \)[/tex]
- [tex]\( P(X = 4) = 0.20 \)[/tex]
- [tex]\( P(X = 5) = 0.27 \)[/tex]

To find the probability of fewer than 3 stars:
[tex]\[ P(X < 3) = P(X = 1) + P(X = 2) \][/tex]

Substituting the given values:
[tex]\[ P(X < 3) = 0.24 + 0.20 = 0.44 \][/tex]

Therefore, the probability that the rating is fewer than 3 stars is:
[tex]\[ P(X < 3) = 0.44 \][/tex]

### Part (b): Probability of 4 or more stars

We need to find the probability that the rating is 4 or more stars. This means we are focusing on the probabilities of ratings 4 stars and 5 stars.

To find the probability of 4 or more stars:
[tex]\[ P(X \geq 4) = P(X = 4) + P(X = 5) \][/tex]

Substituting the given values:
[tex]\[ P(X \geq 4) = 0.20 + 0.27 = 0.47 \][/tex]

Therefore, the probability that the rating is 4 or more stars is:
[tex]\[ P(X \geq 4) = 0.47 \][/tex]

To summarize:
(a) The probability that the rating is fewer than 3 stars is [tex]\( P(X < 3) = 0.44 \)[/tex].
(b) The probability that the rating is 4 or more stars is [tex]\( P(X \geq 4) = 0.47 \)[/tex].