Applied Mathematics – Discrete Random Variables, Financial Math, Optimization
MathematicsSequences & Financial MathAges 17–18
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Sign in to playThree math topics in one simulation, chosen with the topic parameter. Discrete random variable: toss n coins thousands of times and compare the experimental frequency bars with the theoretical probability distribution, computing the expected value and variance. Mathematics in finance: compare simple interest with compound interest A = P(1+r)^n and compute the monthly payment on an amortized loan. Optimization problem: drag the size of the squares cut from the four corners of a sheet of cardboard to fold an open-top box, watch the volume graph V(x) = x(a−2x)², and find the maximum using derivatives.
Lesson: Discrete random variables and their parameters; mathematical applications in finance; using derivatives to optimize
What it shows
Three topics share one page. The number of heads X when n fair coins are tossed is a discrete random variable with P(X = k) = C(n; k)/2ⁿ, expected value n/2 and variance n/4; thousands of simulated tosses bring the experimental frequencies close to this distribution, as the law of large numbers predicts. In finance, simple interest grows linearly, A = P(1 + r·n), while compound interest grows exponentially, A = P(1 + r)ⁿ, and the same model sets the equal monthly payment on a loan. In optimization, cutting squares of side x from a sheet of side a gives a box of volume V(x) = x(a − 2x)², largest at x = a/6.
How to use
Choose the Current topic. For coins, set Number of coins tossed each time, click Toss 200 times or Toss continuously, and compare the red bars with the theoretical outline and the experimental average with E(X). For finance, change Nominal annual interest rate and Number of years deposited or borrowed, switching between View savings account and View amortized loan. For the box, drag on the cardboard to change x, watch V(x), then click Set x = a/6 to check the maximum.
Parameters you can change
- Current topic 1. Discrete random variable, 2. Mathematics in finance, 3. Optimization problem
- Number of coins tossed each time 1–12 coins
- Initial deposit or loan amount 1–500 thousand USD
- Nominal annual interest rate 0.5–20 %/year
- Number of years deposited or borrowed 1–30 years
- Side length of the square cardboard to fold into a box 10–60 cm
Questions to explore
- When 6 coins are tossed, which number of heads is most likely, and what is its probability?
- Why does the gap between compound and simple interest keep growing over time?
- For which values of x can the box be made, and why is V(x) = 0 at both ends of that interval?
