### Probability in Education Lineage Among Ivy League Schools: Harvard, Dartmouth, and Yale#### Problem Context:Historically, Harvard, Dartmouth, and Yale admitted only male students. Assume the following in terms of educational continuity among these institutions:- **Harvard**: 80% of the sons of Harvard alumni attended Harvard, while the remaining 20% attended Yale.- **Yale**: 40% of the sons of Yale alumni attended Yale. The rest were equally divided between Harvard and Dartmouth.- **Dartmouth**: 70% of the sons of Dartmouth alumni attended Dartmouth, 20% attended Harvard, and 10% attended Yale.#### Questions:1. **Find the probability that the grandson of a Harvard alumnus attended Harvard.** To solve this, we need to consider both the son's and the grandson's probabilities of attending Harvard. The calculations involve multiple steps in conditional probability, taking into account the paths through Harvard, Yale, and Dartmouth.2. **Modify the problem by assuming that the son of a Harvard alumnus always attended Harvard. Find the probability that the grandson of a Harvard alumnus attended Harvard.** If it is certain that the son attends Harvard, the problem simplifies as only the probability of the grandson needs to be considered.### Detailed Steps for Calculation:**Question 1:**- Assume the initial probability tree and calculate the following: **For the Harvard alumnus's son:** - Probability (Son attends Harvard) = 0.8 - Probability (Son attends Yale) = 0.2 **For the son who attended Harvard:** - Probability (Grandson attends Harvard given Father attended Harvard) = 0.8 **For the son who attended Yale:** - Probability (Two scenarios for Grandson): - Probability (Grandson attends Harvard given Father attended Yale) = 0.3 * 0 + 0.6 * 0.8 + 0.1 * 0.2 Calculate the combined probability for the grandson attending Harvard by summing the products of the individual path probabilities. **Question 2:**- Given that the son of a Harvard alumnus always attends Harvard: - Probability (Son attends Harvard) = 1.0 Calculate the probability considering only the grandchild: - Probability (Grandson attends Harvard given Father attended Harvard) = 0.8By

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In the Dark Ages, Harvard, Dartmouth, and Yale admitted only male students. Assume that, at that time, 80 percent of the sons of Harvard men went to Harvard and the rest went to Yale, 40 percent of the sons of Yale men went to Yale, and the rest sp1it evenly between Harvard and Dartmouth; and of the sons of Dartmouth men, 70 percent went to Dartmouth, 20 percent to Harvard, and 10 percent to Yale.

In the Dark Ages, Harvard, Dartmouth, and Yale admitted only male students. Assume that, at that time, 80 percent of the sons of Harvard men went to Harvard and the rest went to Yale, 40 percent of the sons of Yale men went to Yale, and the rest sp1it evenly between Harvard and Dartmouth; and of the sons of Dartmouth men, 70 percent went to Dartmouth, 20 percent to Harvard, and 10 percent to Yale.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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### Probability in Education Lineage Among Ivy League Schools: Harvard, Dartmouth, and Yale

#### Problem Context:
Historically, Harvard, Dartmouth, and Yale admitted only male students. Assume the following in terms of educational continuity among these institutions:

- **Harvard**: 80% of the sons of Harvard alumni attended Harvard, while the remaining 20% attended Yale.
- **Yale**: 40% of the sons of Yale alumni attended Yale. The rest were equally divided between Harvard and Dartmouth.
- **Dartmouth**: 70% of the sons of Dartmouth alumni attended Dartmouth, 20% attended Harvard, and 10% attended Yale.

#### Questions:

1. **Find the probability that the grandson of a Harvard alumnus attended Harvard.**

To solve this, we need to consider both the son's and the grandson's probabilities of attending Harvard. The calculations involve multiple steps in conditional probability, taking into account the paths through Harvard, Yale, and Dartmouth.

2. **Modify the problem by assuming that the son of a Harvard alumnus always attended Harvard. Find the probability that the grandson of a Harvard alumnus attended Harvard.**

If it is certain that the son attends Harvard, the problem simplifies as only the probability of the grandson needs to be considered.

### Detailed Steps for Calculation:

**Question 1:**
- Assume the initial probability tree and calculate the following:

**For the Harvard alumnus's son:**
- Probability (Son attends Harvard) = 0.8
- Probability (Son attends Yale) = 0.2

**For the son who attended Harvard:**
- Probability (Grandson attends Harvard given Father attended Harvard) = 0.8

**For the son who attended Yale:**
- Probability (Two scenarios for Grandson):
- Probability (Grandson attends Harvard given Father attended Yale) = 0.3 * 0 + 0.6 * 0.8 + 0.1 * 0.2

Calculate the combined probability for the grandson attending Harvard by summing the products of the individual path probabilities.

**Question 2:**
- Given that the son of a Harvard alumnus always attends Harvard:
- Probability (Son attends Harvard) = 1.0

Calculate the probability considering only the grandchild:
- Probability (Grandson attends Harvard given Father attended Harvard) = 0.8

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