Math Solutions

Compare discrete compounding A = P(1 + r/n)^(nt) with continuous A = Pe^(rt) for $1,000 at 10% for 5 years, as n increases.

A $200,000 mortgage at 4% annual interest for 30 years. Find the monthly payment and show how the principal/interest split evolves over the loan.

Invest $5,000 today at 6% annual interest compounded annually. What is it worth in 20 years?

What is $10,000 received 5 years from now worth today at a 7% discount rate (compounded annually)?

Deposit $200 at the beginning of each month at 6% APR for 20 years. Find the future value and compare to the ordinary-annuity version.

Deposit $200 at the end of each month into an account earning 6% APR for 20 years. Find the future value of this ordinary annuity.

Borrow $5,000 at 8% simple interest for 3 years. Find the total interest owed and compare to compound interest.

Invest $1,000 at 5% annual interest compounded monthly for 10 years. Find the final amount and compare to simple interest.

A bag holds 5 red and 3 blue marbles. Draw 2 marbles. Compare P(both red) with and without replacement. Show both scenarios side by side.

A continuous random variable on [0, 1] has PDF f(x) = 2x. Find P(0.3 < X < 0.7). Show the shaded area under the curve and verify the PDF integrates to 1.

Let X = the number of heads in 3 coin flips. List all 8 outcomes, build the probability distribution table, and graph P(X = k) as a bar chart.

Compute 7! = 5040. Show the chain 7 × 6 × 5 × 4 × 3 × 2 × 1 with a growing bar, and visualize why factorial grows faster than exponential.

Draw two aces in a row without replacement from a 52-card deck. Compute P(2 aces) = 4/52 × 3/51. Show the shrinking deck and contrast with replacement.

Draw one card from a standard deck. Find P(Heart OR Face card). Show why you must subtract P(Heart AND Face) to avoid double counting.

Draw one card from a standard deck. Find P(King OR Queen). Explain why you simply add the two probabilities when the events cannot happen at once.

Weather model: if sunny today, P(sunny tomorrow) = 0.8. If rainy today, P(sunny tomorrow) = 0.4. Draw the state diagram and find the long-run probability of sunny weather.

A standard deck has 13 hearts. Draw 5 cards without replacement. What is P(exactly 2 hearts)? Show the sampling and the full hypergeometric formula.

A free-throw shooter hits 70% of shots. What is the probability that their first make is on the 4th attempt? Show the trial sequence and the full distribution.

You have 4 shirts, 3 pants, and 2 pairs of shoes. How many outfits can you assemble? Show the 4 × 3 × 2 = 24 branching tree.

Choose 3 students from 8 for a committee. Compute C(8,3) = 56. Show why dividing by 3! removes duplicate arrangements.