The Science of Chance
What are the chances that it will rain tomorrow? What's the likelihood of flipping a coin and getting heads? Probability is the branch of mathematics that deals with the chance, or likelihood, that a particular event will happen.
Calculating Simple Probability
The probability of a single event is a ratio comparing the number of favorable outcomes to the total number of possible outcomes.
P(event) = (Number of favorable outcomes) / (Total number of possible outcomes)
Probabilities can be expressed as a fraction, a decimal, or a percentage.
Example: Rolling a standard six-sided die.
Question: What is the probability of rolling a 4?
Favorable outcomes: There is only one way to roll a 4. So, the number is 1.
Total possible outcomes: There are six faces on the die (1, 2, 3, 4, 5, 6). So, the total is 6.
Probability: P(rolling a 4) = 1/6
As a decimal: 1 ÷ 6 ≈ 0.167
As a percent: 0.167 * 100 = 16.7%
The Probability Scale
The probability of any event is always a number between 0 and 1 (or 0% and 100%).
A probability of 0 means the event is impossible. The probability of rolling a 7 on a standard six-sided die is 0.
A probability of 1 means the event is certain. The probability of rolling a number less than 10 on a standard six-sided die is 1 (or 6/6).
Events with a probability of 0.5 (or 1/2 or 50%) are equally likely to happen or not happen. Flipping heads on a fair coin has a probability of 0.5.
Theoretical vs. Experimental Probability
There are two main ways we can think about probability.
Theoretical Probability: This is what we expect to happen based on mathematics and theory. We calculate it using the formula above without actually doing the experiment. We know that the theoretical probability of flipping heads is 1/2.
Experimental Probability: This is what actually happens when we perform an experiment. It is calculated by repeating an experiment and observing the results.
P(event) = (Number of times the event occurred) / (Total number of trials)
Example: You flip a coin 10 times and get 4 heads.
The experimental probability of getting heads in your experiment is 4/10 (or 2/5).
The Law of Large Numbers: The more times you repeat an experiment (the more trials you do), the closer the experimental probability will get to the theoretical probability. If you flip a coin 1000 times, your result will be much closer to 50% heads than it was in just 10 flips.