Problem Analysis
We are given a two-way table showing the distribution of 120 students based on their handedness (left-handed or right-handed) and the subjects they take (French or Spanish). The task is to solve four probability questions using the provided data.
#### Two-Way Table:
| | Left handed | Right handed | Total |
|-----------|-------------|--------------|-------|
| French | 20 | 40 | 60 |
| Spanish | 18 | 42 | 60 |
| Total | 38 | 82 | 120 |
Questions to Solve:
1.
Find the probability of a student taking French or Spanish.
2.
Find the probability of a student taking French and is right-handed.
3.
Given that a student is right-handed, what is the probability they take French?
4.
Given that a student is right-handed, what is the probability they take Spanish?
---
Solution:
####
1. Find the probability of a student taking French or Spanish.
- From the table, we see that all students are either taking French or Spanish (or both). The total number of students is 120.
- Since every student is accounted for in either French or Spanish, the probability of a student taking French or Spanish is:
$$
P(\text{French or Spanish}) = \frac{\text{Total number of students}}{\text{Total number of students}} = \frac{120}{120} = 1
$$
Answer:
$$
\boxed{1}
$$
---
####
2. Find the probability of a student taking French and is right-handed.
- From the table, the number of students who are right-handed and taking French is 40.
- The total number of students is 120.
- The probability is:
$$
P(\text{French and Right-handed}) = \frac{\text{Number of students taking French and Right-handed}}{\text{Total number of students}} = \frac{40}{120} = \frac{1}{3}
$$
Answer:
$$
\boxed{\frac{1}{3}}
$$
---
####
3. Given that a student is right-handed, what is the probability they take French?
- We need to find the conditional probability \( P(\text{French} \mid \text{Right-handed}) \).
- From the table, the number of right-handed students is 82.
- Among these right-handed students, 40 are taking French.
- The conditional probability is:
$$
P(\text{French} \mid \text{Right-handed}) = \frac{\text{Number of right-handed students taking French}}{\text{Total number of right-handed students}} = \frac{40}{82} = \frac{20}{41}
$$
Answer:
$$
\boxed{\frac{20}{41}}
$$
---
####
4. Given that a student is right-handed, what is the probability they take Spanish?
- We need to find the conditional probability \( P(\text{Spanish} \mid \text{Right-handed}) \).
- From the table, the number of right-handed students is 82.
- Among these right-handed students, 42 are taking Spanish.
- The conditional probability is:
$$
P(\text{Spanish} \mid \text{Right-handed}) = \frac{\text{Number of right-handed students taking Spanish}}{\text{Total number of right-handed students}} = \frac{42}{82} = \frac{21}{41}
$$
Answer:
$$
\boxed{\frac{21}{41}}
$$
---
Final Answers:
1. \( \boxed{1} \)
2. \( \boxed{\frac{1}{3}} \)
3. \( \boxed{\frac{20}{41}} \)
4. \( \boxed{\frac{21}{41}} \)
Parent Tip: Review the logic above to help your child master the concept of two way table probability worksheet.
