To find the missing probability, we can follow these detailed steps:
### Step-by-Step Solution:
1. Identify Given Data:
The table shows the number of students (X) and their corresponding probabilities [tex]\( P(X) \)[/tex]. The given values are:
- [tex]\( P(22) = 0.1 \)[/tex]
- [tex]\( P(23) = 0.14 \)[/tex]
- [tex]\( P(24) = 0.17 \)[/tex]
- [tex]\( P(25) = \)[/tex] (missing)
- [tex]\( P(26) = 0.12 \)[/tex]
2. Sum of Given Probabilities:
Let's sum the provided probabilities:
[tex]\[
P(22) + P(23) + P(24) + P(26) = 0.1 + 0.14 + 0.17 + 0.12
\][/tex]
Calculate this sum:
[tex]\[
0.1 + 0.14 + 0.17 + 0.12 = 0.53
\][/tex]
3. Determine Missing Probability:
The probabilities in a probability distribution must sum to 1. Therefore, the missing probability is:
[tex]\[
P(25) = 1 - 0.53
\][/tex]
Calculate the missing value:
[tex]\[
P(25) = 0.47
\][/tex]
4. Fill the Missing Value in the Table:
Now, we can update the table with the missing probability value.
\begin{tabular}{|c|c|}
\hline X & [tex]$P(X)$[/tex] \\
\hline 22 & 0.1 \\
\hline 23 & 0.14 \\
\hline 24 & 0.17 \\
\hline 25 & 0.47 \\
\hline 26 & 0.12 \\
\hline
\end{tabular}
Therefore, the missing value is [tex]\( P(25) = 0.47 \)[/tex], and the complete probability distribution is:
[tex]\[
P(X) = [0.1, 0.14, 0.17, 0.47, 0.12]
\][/tex]
