Lesson 10-4: Compare Probabilities of Simple Events

Property

When the probability of an event is known, or can be determined through analysis where all outcomes are equally likely, the theoretical probability is:

Examples

• The theoretical probability of rolling a 2 on a six-sided die is $\frac{1}{6}$. If you roll it 12 times and get a 2 three times, the experimental probability is $\frac{3}{12}$ or $\frac{1}{4}$.
• A bag has 4 red and 6 blue marbles. The theoretical probability of drawing red is $\frac{4}{10} = \frac{2}{5}$. After drawing and replacing 20 times, you draw red 9 times. The experimental probability is $\frac{9}{20}$.
• A spinner has 4 equal sections. The theoretical probability of landing on 'A' is $\frac{1}{4}$. After 60 spins, it lands on 'A' 12 times. The experimental probability is $\frac{12}{60} = \frac{1}{5}$.

Explanation

Theoretical probability is what should happen based on pure math, like a $\frac{1}{2}$ chance of heads. Experimental probability is what actually happens when you run the experiment, like getting 47 heads in 100 flips.

Property

When comparing theoretical probability ($P_{\text{theo}}$) and experimental probability ($P_{\text{exp}}$), significant differences may indicate that the theoretical model is inappropriate.

Examples

• A coin is flipped $100$ times and lands on heads $85$ times ($P_{\text{exp}} = 0.85$). Since the theoretical probability is $P_{\text{theo}} = 0.50$, the coin is likely biased.
• A spinner has $4$ equal sections. After $1000$ spins, it lands on red $245$ times ($P_{\text{exp}} = 0.245$). This is very close to $P_{\text{theo}} = 0.25$, suggesting the spinner is fair.
• A basketball player makes $80$ out of $100$ free throws ($P_{\text{exp}} = 0.80$). A theoretical model assuming a $50\%$ chance of making a shot ($P_{\text{theo}} = 0.50$) is inappropriate because shooting is a skill, not a purely random event.

Explanation

Comparing theoretical and experimental probabilities helps us determine if a game, object, or process is truly fair and random. Small differences between the two are normal and expected due to chance. However, if the experimental probability is vastly different from the theoretical probability, especially after many trials, it suggests the theoretical model is flawed. This usually means the object is biased, the process is not truly random, or human skill is involved.

Property

The Law of Large Numbers states that as the number of trials in a probability experiment increases, the experimental probability of an event approaches its theoretical probability.