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A candle shop owner collected data on customers' favorite scents, as shown in the table.

\begin{tabular}{|c|c|c|c|c|}
\hline
Lavender & Vanilla & Cinnamon & Floral & Coconut \\
\hline
0.31 & 0.18 & 0.23 & [tex]$?$[/tex] & [tex]$?$[/tex] \\
\hline
\end{tabular}

A customer is randomly selected. If the probabilities of a customer choosing floral and coconut as their favorite scents are the same, what is the value of the missing probabilities?

A. 0.10
B. 0.14
C. 0.20
D. 0.28


Sagot :

Let's solve this problem step-by-step by analyzing the given data and using the condition that the probabilities for floral and coconut are equal.

1. First, let's write down the given probabilities:
- Lavender: 0.31
- Vanilla: 0.18
- Cinnamon: 0.23
- Floral: ?
- Coconut: ?

2. The sum of all probabilities must equal 1 since these represent all possible outcomes:
[tex]\[ P(\text{Lavender}) + P(\text{Vanilla}) + P(\text{Cinnamon}) + P(\text{Floral}) + P(\text{Coconut}) = 1 \][/tex]

3. Filling in the known values, we have:
[tex]\[ 0.31 + 0.18 + 0.23 + P(\text{Floral}) + P(\text{Coconut}) = 1 \][/tex]

4. Calculating the sum of the given probabilities:
[tex]\[ 0.31 + 0.18 + 0.23 = 0.72 \][/tex]

5. Therefore, we can update our equation:
[tex]\[ 0.72 + P(\text{Floral}) + P(\text{Coconut}) = 1 \][/tex]

6. To find the combined probability of floral and coconut, we subtract 0.72 from 1:
[tex]\[ 1 - 0.72 = 0.28 \][/tex]
So,
[tex]\[ P(\text{Floral}) + P(\text{Coconut}) = 0.28 \][/tex]

7. Given the condition that the probabilities for floral and coconut are the same:
[tex]\[ P(\text{Floral}) = P(\text{Coconut}) \][/tex]

8. Let’s denote \( P(\text{Floral}) \) and \( P(\text{Coconut}) \) as \( f \). Thus:
[tex]\[ f + f = 0.28 \][/tex]

9. Solving for \( f \):
[tex]\[ 2f = 0.28 \][/tex]
[tex]\[ f = \frac{0.28}{2} \][/tex]
[tex]\[ f = 0.14 \][/tex]

Therefore, the missing probabilities are:
- Floral: 0.14
- Coconut: 0.14

Among the provided options, the correct answer is [tex]\( \boxed{0.14} \)[/tex].