A Super Mega Bowl guide
Probability is just a way of putting a number on how likely something is. Once you can read that number and combine a few simple rules, most of the confusion around chance falls away. Here is the whole foundation in one place.
A probability is a number between 0 and 1 that measures how likely an event is. A probability of 0 means it never happens, 1 means it always happens, and 0.5 means it happens half the time. For any event with equally likely outcomes, you find the probability by dividing the number of ways it can happen by the total number of possible outcomes.
A fair coin has two outcomes, so heads has a probability of 1 out of 2, which is 0.5. A fair six-sided die has six outcomes, so rolling a 4 has a probability of 1 out of 6, about 0.167.
The same likelihood can be written three ways, and it helps to move between them.
| Form | Rolling a 4 on a die |
|---|---|
| Fraction | 1/6 |
| Percent | about 16.7 percent |
| Odds | 1 to 5 (one way to win, five ways to lose) |
Odds and probability are not the same thing, and mixing them up is a common error. Odds of 1 to 5 means one favorable outcome against five unfavorable ones, which is a probability of 1 out of 6, not 1 out of 5.
Theoretical probability is what the math predicts: a fair coin should land heads half the time. Experimental probability is what you actually observe: flip a coin twenty times and you might see heads 12 times, an experimental probability of 0.6. The two do not have to match in a small sample. They converge as the number of trials grows, which is the law of large numbers.
Two events are independent when one does not affect the other. Coin flips, dice rolls, and spins are independent: the coin has no memory, so a run of heads does not change the next flip. Believing it does is the gambler's fallacy.
Events are dependent when one changes the odds of the next. Drawing cards from a deck without putting them back is dependent, because each card removed changes what is left. Knowing which kind you are dealing with tells you whether the odds stay fixed or shift as you go.
Two rules cover most everyday problems.
A useful shortcut is the complement. The probability that something does not happen is 1 minus the probability that it does. The chance of at least one head in three flips is easiest to find as 1 minus the chance of no heads, which is 1 minus 0.5 times 0.5 times 0.5, or about 88 percent.
Numbers on a page are easy to forget. Flip a batch of coins or roll a handful of dice, watch the results pile up, and compare what you see against what the theory predicts. That gap between theory and experiment, and the way it closes with more trials, is the heart of probability.
Start experimenting → Flip coins and roll dice, then check the running tallies against the odds you just learned.Probability is a number from 0 to 1 that you can read as a fraction, a percent, or odds. Theory predicts the long run, experiment shows the short run, and a couple of rules for AND and OR handle most real questions. Everything else in probability builds on these basics.