Probability Models and Simulations

A probability model is a mathematical way to represent a random event. It helps us understand and predict the likelihood of different outcomes.

There are two main types of probability models:

• Uniform Probability Model: Every outcome has an equal chance of occurring. For example, rolling a fair six-sided die or flipping a coin.
• Non-uniform Probability Model: Outcomes have different chances of occurring. For example, spinning a spinner where half the circle is red, a quarter is blue, and a quarter is green.

Theoretical vs. Experimental Probability

To truly understand probability, we have to look at the difference between what math tells us should happen and what actually happens.

• Theoretical Probability is what we expect to happen based on the math. $P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$
• Experimental Probability is what actually happens when we run an experiment. $P(\text{event}) = \frac{\text{number of times the event occurs}}{\text{total number of trials}}$

Example: Imagine you roll a six-sided die 100 times. The theoretical probability of rolling a 3 is exactly $\frac{1}{6}$, or about $16.7\%$. However, after rolling the die 100 times, you might actually roll a 3 exactly 18 times. Your experimental probability is $\frac{18}{100} = 18\%$.

Note: As you increase the number of trials (like rolling the die 1,000 times instead of 100), your experimental probability will usually get closer and closer to the theoretical probability!

What is a Simulation?

Sometimes, it is too difficult, expensive, or time-consuming to perform an actual experiment. A simulation is a way to model random events using tools like dice, coins, spinners, or random number generators to mimic the real-world situation.

Designing a Simulation

Let's say a basketball player makes $75\%$ of her free throws. How can we design a simulation to estimate the probability of her making 3 shots in a row?

1. Choose a tool: Since $75\% = \frac{3}{4}$, we need a tool with 4 equal outcomes. We can use a spinner divided into 4 equal sections, or a random number generator picking numbers 1 through 4.
2. Assign outcomes: Let the numbers 1, 2, and 3 represent a "made shot" (since that's 3 out of 4). Let the number 4 represent a "missed shot".
3. Run trials: Generate 3 random numbers to represent 3 consecutive shots. For example, getting 2, 1, 4 means she made the first two and missed the third.
4. Record and analyze: Repeat this 3-shot trial many times (e.g., 50 times) to find the experimental probability of her making all 3 shots.

Using Random Numbers

Computers and calculators are great for simulations. If you want to simulate a lottery drawing where 3 winning numbers are chosen from 1 to 50, you don't need to buy thousands of tickets. Instead, you can use a computer's random number generator to pick 3 distinct numbers between 1 and 50. You can program the computer to run this simulation 10,000 times in just a few seconds to estimate your chances of winning!