Q1. What is the probability that at least one of the weatherman's 320 forecasts is incorrect, given the probability of being completely wrong is 0.015?

Background

Topic: Probability of Complementary Events

This question tests your understanding of the probability of at least one event occurring in repeated independent trials, using the complement rule.

Key Terms and Formulas

• Complement Rule:

• Probability of none being incorrect:

• Where is the probability of being wrong on a single day, and is the number of days.

Step-by-Step Guidance

1. Identify the probability of being wrong on a single day: .

2. Calculate the probability of being correct on a single day: .

3. Find the probability that the weatherman is correct every day for 320 days: .

4. Apply the complement rule to find the probability that at least one forecast is incorrect: .

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Q2. If a student is selected at random from the survey, 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 data from a contingency table.

Key Terms and Formulas

• Probability:

Step-by-Step Guidance

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

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

3. Set up the probability formula: .

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

Background

Topic: Conditional Probability

This question tests your ability to calculate conditional probabilities using a two-way table.

Key Terms and Formulas

• Conditional Probability:

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: .

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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 calculate the probability of one event given another, using a two-way table.

Key Terms and Formulas

• Conditional Probability:

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: .

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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 A and B are independent if .

• Compare and .

Step-by-Step Guidance

1. Calculate from the total table.

2. Calculate from the girls row.

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

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