Probability
Probability measures how likely an event is to occur, expressed as a number from 0 (impossible) to 1 (certain).
Basic Probability
P(event) = favorable outcomes / total outcomes
Rolling a 3 on a fair die: P = 1/6 ≈ 0.167 ≈ 16.7%
Drawing a red card from a deck: P = 26/52 = 1/2 = 50%
Theoretical vs Experimental
Theoretical: calculated from math (fair coins, dice).
Experimental: based on actual trials. As trials increase, experimental approaches theoretical.
Complementary Events
P(event NOT happening) = 1 − P(event)
If P(rain) = 0.3, then P(no rain) = 0.7
Compound Events
Independent events (one doesn't affect the other):
P(A and B) = P(A) × P(B)
Flipping heads twice: (1/2) × (1/2) = 1/4
Step by Step: Setting Up a Probability
Almost every probability question comes down to two counts: how many outcomes you want, and how many outcomes exist in total.
A bag holds 4 red, 6 blue, and 2 green marbles. What is P(blue)?
- Count the favorable outcomes: 6 blue marbles.
- Count all outcomes:
4 + 6 + 2 = 12marbles. - Divide:
6/12 = 1/2 = 50%.
Sanity check: the answer is between 0 and 1, and blue is the most common colour, so a probability at or above one third makes sense.
Dependent Events
Independent events do not affect each other. Dependent events do, and the difference changes the arithmetic.
Drawing two marbles without replacement
From the same bag, what is the probability of drawing two blues in a row if you keep the first marble?
First draw: 6/12. Now only 5 blues remain out of 11 marbles, so the second draw is 5/11.
P(blue, then blue) = 6/12 × 5/11 = 30/132 ≈ 0.227
If you had put the first marble back, both draws would be 6/12 and the events would be independent.
"And" Versus "Or"
- And means both happen, so you multiply. Probabilities get smaller.
- Or means at least one happens, so you add — then subtract the overlap if the events can happen together.
Rolling one die: P(even or greater than 4)
Even: 2, 4, 6 → 3/6. Greater than 4: 5, 6 → 2/6.
The number 6 is in both lists, so subtract it once: 3/6 + 2/6 − 1/6 = 4/6 = 2/3.
Common Mistakes
- Forgetting that the total includes the favorable outcomes. In a bag of 4 red and 6 blue, P(red) is
4/10, not4/6. - The gambler's fallacy. A coin that landed heads five times running is still exactly 50/50 on the next flip. Coins have no memory.
- Adding when the question says "and". Two things both happening is rarer than either one alone, so the answer must get smaller.
- Reporting a probability above 1. If that happens, you have added when you should have multiplied, or double-counted an overlap.
Where You Will Use This
- Weather forecasts — a 30% chance of rain is a probability statement.
- Medical test results, where the chance of a false positive matters as much as the result.
- Insurance pricing, which is probability applied to risk.
- Games and sports statistics, from batting averages to win projections.
Summary
- Probability is favorable outcomes divided by total outcomes, always between 0 and 1.
- Theoretical probability comes from reasoning; experimental comes from trials, and they converge as trials increase.
- The complement rule,
1 − P(event), is often the fastest route to an answer. - Multiply for "and", add for "or", and subtract any overlap.
- Check whether events are independent before you multiply.
FAQ
Can probability be greater than 1? No. Values range from 0 to 1 only.
What does P = 0.5 mean? The event is equally likely to happen or not.
What is the fastest way to handle "at least one"? Use the complement. P(at least one) is 1 − P(none), and "none" is usually far easier to count.
Why does experimental probability not match theoretical? Random variation. Flip a coin ten times and you might get seven heads; flip it ten thousand times and you will land very close to half.
Can two events be both independent and mutually exclusive? Not unless one is impossible. Mutually exclusive events affect each other completely — if one happens, the other cannot.
How do I convert a probability to a percentage? Multiply by 100. A probability of 0.25 is a 25% chance.
Quick Quiz
Test what you just learned. Choose the best answer for each question.