Introductory Statistics Study Guide: Conditional Probability & Independence

Q1. What is the probability that at least one of the weatherman's 320 forecasts is incorrect?

Background

Topic: Probability of Complementary Events

This question tests your understanding of the complement rule in probability, specifically for repeated independent events.

Key Terms and Formulas

• Complement Rule: $P( ext{at least one error}) = 1 - P( ext{no errors})$

• Probability of no errors in 320 forecasts: $P( ext{no errors}) = (1 - p)^{320}$, where $p$ is the probability of being wrong.

Step-by-Step Guidance

1. Identify the probability of being wrong for a single forecast: $p = 0.015$.

2. Calculate the probability of being correct for a single forecast: $1 - p = 0.985$.

3. Find the probability that all 320 forecasts are correct: $(0.985)^{320}$.

4. Apply the complement rule to find the probability that at least one forecast is incorrect: $1 - (0.985)^{320}$.

Try solving on your own before revealing the answer!

Final Answer: 0.99 (or 99%)

$P( ext{at least one error}) = 1 - (0.985)^{320} \approx 0.99$

With so many forecasts, the probability that at least one is incorrect is extremely high.

Q2. If a student is selected at random, what is the probability the student wants to be Popular?

Background

Topic: Basic Probability from a Two-Way Table

This question tests your ability to calculate simple probabilities using frequency tables.

Key Terms and Formulas

• Probability: $P( ext{Popular}) = \frac{\text{Number who want to be popular}}{\text{Total number of students}}$

Step-by-Step Guidance

1. Find the total number of students: Add all values in the table.

2. Find the number of students who want to be popular: Add the "Be Popular" values for boys and girls.

3. Set up the probability formula: $P( ext{Popular}) = \frac{\text{Number who want to be popular}}{\text{Total number of students}}$

Try solving on your own before revealing the answer!

Final Answer: 0.295 (or 29.5%)

$P( ext{Popular}) = \frac{50 + 91}{478} = \frac{141}{478} \approx 0.295$

141 out of 478 students want to be popular.

Q3. What is the probability the student wants to be popular given it is a girl?

Background

Topic: Conditional Probability

This question tests your understanding of conditional probability using a two-way table.

Key Terms and Formulas

• Conditional Probability: $P( ext{Popular} | ext{Girl}) = \frac{\text{Number of girls who want to be popular}}{\text{Total number of girls}}$

Step-by-Step Guidance

1. Find the total number of girls: Add all values in the "Girls" row.

2. Find the number of girls who want to be popular: Use the "Be Popular" value for girls.

3. Set up the conditional probability formula: $P( ext{Popular} | ext{Girl}) = \frac{\text{Number of girls who want to be popular}}{\text{Total number of girls}}$

Try solving on your own before revealing the answer!

Final Answer: 0.385 (or 38.5%)

$P( ext{Popular} | ext{Girl}) = \frac{91}{130 + 91 + 30} = \frac{91}{251} \approx 0.385$

About 38.5% of girls surveyed want to be popular.

Q4. What is the probability the student is a girl given we selected a student wanting to be popular?

Background

Topic: Conditional Probability (Reverse)

This question tests your ability to reverse the conditioning in probability using a two-way table.

Key Terms and Formulas

• Conditional Probability: $P( ext{Girl} | ext{Popular}) = \frac{\text{Number of girls who want to be popular}}{\text{Total number who want to be popular}}$

Step-by-Step Guidance

1. Find the total number of students who want to be popular: Add "Be Popular" values for boys and girls.

2. Find the number of girls who want to be popular: Use the "Be Popular" value for girls.

3. Set up the conditional probability formula: $P( ext{Girl} | ext{Popular}) = \frac{\text{Number of girls who want to be popular}}{\text{Total number who want to be popular}}$

Try solving on your own before revealing the answer!

Final Answer: 0.646 (or 64.6%)

$P( ext{Girl} | ext{Popular}) = \frac{91}{141} \approx 0.646$

About 64.6% of students who want to be popular are girls.

Q5. Are being Popular and being a Girl independent?

Background

Topic: Independence of Events

This question tests your understanding of independence in probability, using conditional and marginal probabilities.

Key Terms and Formulas

• Events are independent if $P( ext{Popular} | ext{Girl}) = P( ext{Popular})$

Step-by-Step Guidance

1. Recall the probability a student wants to be popular: $P( ext{Popular})$

2. Recall the probability a student wants to be popular given they are a girl: $P( ext{Popular} | ext{Girl})$

3. Compare these two probabilities. If they are equal, the events are independent; if not, they are dependent.

Try solving on your own before revealing the answer!

Final Answer: Not independent

$P( ext{Popular}) \approx 0.295$ and $P( ext{Popular} | ext{Girl}) \approx 0.385$ are not equal, so the events are not independent.

Being a girl affects the probability of wanting to be popular.