Math Solutions

How many ways can 5 runners finish 1st, 2nd, and 3rd? Compute P(5,3) = 60. Show the arrangement tree and explain when order matters.

A coin flip and a die roll happen together. Show P(heads AND 6) = P(heads) × P(6) = 1/12. Then compare to drawing two cards without replacement to see how dependence changes the rule.

A bag has 4 red, 3 blue balls. Draw one red, then without replacement, P(second is red)?

P(at least one head in 3 coin flips) = 1 - P(all tails). Show the complement.

What is P(sum of 7 with two dice)? Show the 6×6 outcome grid with favorable outcomes highlighted.

Group A: mean=85, s=5, n=20. Group B: mean=80, s=6, n=25. Are they significantly different?

In a poll of 1000 people, 52% support a proposal. Find the margin of error at 95% confidence.

Population σ = 20, sample size n = 64. Calculate SE = σ/√n = 2.5.

Course grades: Homework 85 (20%), Midterm 78 (30%), Final 92 (50%). Find weighted average.

Take 100 random samples of size 25 from a population with μ=50, σ=10. Plot sample means.

Calculate IQR for {1, 3, 5, 7, 9, 11, 13, 15, 17}. Show IQR as the middle 50%.

Find min, Q1, median, Q3, max for {12, 15, 18, 22, 25, 28, 30, 35, 40}. Draw the box plot.

Plot height vs weight for 15 students. Show positive correlation trend line with equation.

Dataset: {3, 5, 7, 8, 9, 10, 45}. Determine if 45 is an outlier using the 1.5×IQR rule.

If p = 0.03 and α = 0.05, show the p-value as shaded tail area on the test distribution.

Visualize α (false positive) and β (false negative) on overlapping null/alternative distributions.

Three teaching methods give scores: A={85,90,88}, B={78,82,80}, C={92,95,88}. Are means different?

Roll a die 10,000 times. Plot sample means of size 30. Show convergence to normal.

Find the 25th, 50th, and 75th percentiles of 20 exam scores. Show position on ordered data.

Create a stem-and-leaf plot for {23, 25, 28, 31, 34, 37, 42, 45, 48, 51}.