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solve part c: under 61 and part d: over 30 and under 51

To solve parts (c) and (d), let's calculate the probability based on the provided age distribution of CEOs.

### Age Distribution:
- 21-30: 1 CEO
- 31-40: 8 CEOs
- 41-50: 27 CEOs
- 51-60: 29 CEOs
- 61-70: 24 CEOs
- 71-up: 11 CEOs

#### Total number of CEOs:
\[
1 + 8 + 27 + 29 + 24 + 11 = 100 \text{ CEOs}
\]

### Part (c): Probability of a CEO being under 61.
This includes the categories for ages 21-30, 31-40, 41-50, and 51-60. So we sum the frequencies for these categories:
\[
1 + 8 + 27 + 29 = 65
\]
Thus, the probability is:
\[
P(\text{under 61}) = \frac{65}{100} = 0.65
\]

### Part (d): Probability of a CEO being over 30 and under 51.
This includes the categories for ages 31-40 and 41-50. So we sum the frequencies for these categories:
\[
8 + 27 = 35
\]
Thus, the probability is:
\[
P(30 < \text{age} < 51) = \frac{35}{100} = 0.35
\]

Let me know if you need further clarification or any additional details!

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Here are 5 related questions to deepen your understanding:
1. What is the probability of a CEO being between 41 and 70?
2. How would the probabilities change if more CEOs are added to the 31-40 category?
3. Can you calculate the probability of a CEO being exactly 71 years or older?
4. If you were to randomly select 2 CEOs, what is the probability that both are over 60?
5. How would the distribution affect decision-making in hiring CEOs of different age groups?

**Tip**: Always check that the sum of your probabilities adds up to 1 as a verification step.

Learn how to calculate the probability of a CEO being under 61 years of age or between 31 and 50 years old using basic probability formulas. This problem involves analyzing the frequency distribution of CEO ages and applying probability theory. Suitable for Grades 9-12.

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