Math Solutions

Solve y=2x+1 and y=−x+7 by graphing. The intersection point (2, 5) is the solution.

Transform x²+6x+5 into (x+3)²−4. Show the geometric square-building process that makes this algebraic identity visual.

Show how b²−4ac determines whether ax²+bx+c=0 has 2, 1, or 0 real roots. Drag a, b, c sliders and watch the parabola cross, touch, or miss the x-axis.

3 doors: 1 car, 2 goats. You pick a door. Monty opens a goat door. Should you switch? Simulate 1000 games to prove switching wins 2/3 of the time.

Roll a die 120 times: observed {18,22,16,20,24,20}. Is it fair? Expected: 20 each. Compute χ² and p-value.

Find the standard deviation of {4, 8, 6, 5, 3}. Show each point's distance from the mean.

Roll a fair die repeatedly. Show how the running average converges to 3.5 as the number of rolls increases.

A game costs $5 to play. You win $20 with probability 0.2 and $0 otherwise. Compute E(X) = Σ xᵢ pᵢ and decide if it is worth playing.

A coin is flipped 100 times and lands heads 60 times. Is it fair? Compute the p-value and show the rejection region.

Take 50 random samples of size 30 from a population. Show how ~95% of 95% confidence intervals capture the true mean.

From 5 students, choose 3 for a committee (combination: C(5,3) = 10) vs arrange 3 in a line (permutation: P(5,3) = 60). Show why order matters.

A disease affects 1% of people. Test is 99% accurate. If you test positive, what's P(disease)?

Flip two coins. Show all outcomes (HH, HT, TH, TT) with probabilities on branches. Extend to conditional probability.

Dataset: {2, 3, 4, 5, 5, 6, 100}. See how the outlier 100 pulls the mean far from the median.

Show how changing the bin width of a histogram dramatically changes the visual shape of the same dataset.

Create a box-and-whisker plot from data. Show median, Q1, Q3, IQR, whiskers, and outliers.

Show scatter plots with different correlation values r = 0.95, 0.5, 0, −0.8. Learn to estimate correlation strength from the visual pattern.

Fit a line to data: (1,2), (2,4), (3,5), (4,4), (5,5). Show residual squares minimized.

A call center receives λ = 5 calls per hour on average. Show P(k calls) = e^(−λ) λ^k / k! for k = 0, 1, 2, ...

Flip a coin n = 20 times with probability p of heads. Show P(k heads) = C(n,k) p^k (1−p)^(n−k) for k = 0 to n as a bar chart.