To find the missing probability in the given probability distribution, let's proceed step-by-step to solve the problem.
1. Identify the given probabilities:
- Probability for the interval [tex]\(0-10\)[/tex]: [tex]\(0.105\)[/tex]
- Probability for the interval [tex]\(11-20\)[/tex]: [tex]\(0.255\)[/tex]
- Probability for the interval [tex]\(31-40\)[/tex]: [tex]\(0.301\)[/tex]
- Probability for the interval [tex]\(41-50\)[/tex]: [tex]\(0.214\)[/tex]
2. Recall the property of a probability distribution:
The total probability of all intervals in a probability distribution must sum up to [tex]\(1\)[/tex].
3. Sum the known probabilities:
[tex]\[
0.105 + 0.255 + 0.301 + 0.214
\][/tex]
Adding them step by step:
[tex]\[
0.105 + 0.255 = 0.360
\][/tex]
[tex]\[
0.360 + 0.301 = 0.661
\][/tex]
[tex]\[
0.661 + 0.214 = 0.875
\][/tex]
4. Determine the missing probability:
Since the total probability is [tex]\(1\)[/tex], the missing probability [tex]\(P_{21-30}\)[/tex] is:
[tex]\[
1 - 0.875 = 0.125
\][/tex]
Therefore, the missing probability for the interval [tex]\(21-30\)[/tex] is [tex]\(\boxed{0.125}\)[/tex]. This corresponds to option C in the given choices.
