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].
