Lesson 5: Designing Simulations

Property

A coin toss can be used to simulate an event where there are two equally likely outcomes. Heads (H) can represent one outcome and Tails (T) can represent the other. The theoretical probability of each outcome is $\frac{1}{2}$.

Examples

• To simulate the probability of a baby being a boy or a girl, you can flip a coin. Let Heads represent a boy and Tails represent a girl.
• To simulate the probability of guessing the correct answer on a true/false question, you can flip a coin. Let Heads represent a correct answer and Tails represent an incorrect answer.
• To simulate the probability that a basketball player who makes 50% of their free throws will make their next shot, you can flip a coin. Let Heads represent a made shot and Tails represent a missed shot.

Explanation

A coin is a simple tool for simulating events with two equally likely outcomes. By assigning one outcome to Heads and the other to Tails, you can model situations with a 50% chance of success. For example, flipping a coin can simulate guessing on a true/false test or predicting the gender of a baby. Running multiple trials by flipping the coin many times allows you to estimate the probability of a compound event.

Property

To simulate an event with a spinner, the area of each sector must be proportional to the probability of the outcome it represents. The central angle for a sector representing an outcome with probability $P$ is calculated as:

Examples

• A basketball player makes 60% of their free throws. To simulate a free throw attempt, a spinner would have a "Make" section with an angle of $0.60 \times 360^\circ = 216^\circ$ and a "Miss" section with an angle of $0.40 \times 360^\circ = 144^\circ$.
• A factory finds that 1 in 20 products is defective. To simulate checking a product, a spinner would have a "Defective" section with an angle of $\frac{1}{20} \times 360^\circ = 18^\circ$ and a "Not Defective" section with an angle of $\frac{19}{20} \times 360^\circ = 342^\circ$.

Explanation

A spinner is a useful tool for modeling the probability of a real-world event. By dividing the spinner into sections, you can create a model where each section's size corresponds to the probability of a specific outcome. The central angle of each sector is determined by multiplying the event's probability by $360^\circ$. Spinning the spinner multiple times allows you to conduct trials and estimate probabilities for more complex, compound events.

Property

Random number generators can simulate probability experiments by assigning number ranges to outcomes based on their theoretical probabilities. For an event with probability $P(A) = \frac{a}{b}$, assign the outcome to occur when the random number falls within a proportional range.

Examples