Probability is a fraction that tells you how likely something is to happen
Probability is a number between 0 and 1 that describes the chance of an event occurring. Zero means it will never happen. One means it will definitely happen. Everything in between is a maybe—the closer to 1, the more likely.
The basic formula is straightforward: divide the number of outcomes you want by the total number of possible outcomes. If you flip a coin, there are two possible outcomes (heads or tails). If you want heads, that's one outcome you want. So the probability is 1 ÷ 2 = 0.5, or 50 percent. That's it. Everything else is explore this same idea to different situations.
Key Takeaways
- Probability = (number of outcomes you want) ÷ (total number of possible outcomes).
- A probability of 0 means something will never happen; a probability of 1 means it will definitely happen.
- You can express probability as a fraction (1/2), a decimal (0.5), or a percentage (50%).
- When events are independent—like rolling two dice—multiply the probabilities together to find the chance both happen.
- When events depend on each other, the probability of the second event changes based on what happened first.
The basic probability formula with a straightforward example
The formula works like this: Probability = Favorable Outcomes ÷ Total Possible Outcomes.
Say you have a bag with 3 red marbles and 7 blue marbles. You reach in and pull one marble without looking. The total number of possible outcomes is 10 (you could pull any of the 10 marbles). The number of favorable outcomes—pulling red—is 3. So the probability of pulling red is 3 ÷ 10 = 0.3, or 30 percent.
If instead you wanted the probability of pulling blue, you'd count 7 favorable outcomes. So 7 ÷ 10 = 0.7, or 70 percent. Notice that 30 percent plus 70 percent equals 100 percent. That's always true: all the probabilities of all possible outcomes add up to 1 (or 100 percent).
Converting between fractions, decimals, and percentages
Probability can be written three ways, and they all mean the same thing. A fraction like 1/4 is the raw ratio. A decimal like 0.25 is what you get when you divide the top by the bottom. A percentage like 25% is the decimal multiplied by 100.
To convert a fraction to a decimal, divide the top number by the bottom. 1 ÷ 4 = 0.25. To convert a decimal to a percentage, multiply by 100. 0.25 × 100 = 25%. To go backward from a percentage to a decimal, divide by 100. 25 ÷ 100 = 0.25. To go from a decimal to a fraction, count how many decimal places you have. 0.25 has two decimal places, so put it over 100: 25/100, which simplifies to 1/4.
In practice, use whichever form makes sense for your situation. Percentages are easiest to talk about ("there's a 25% chance"). Decimals are easiest for math. Fractions are exact.
Independent events: rolling dice or flipping coins multiple times
An independent event is one where the outcome doesn't change the odds of the next event. Rolling a die, getting a 3, and then rolling again—the second roll is not affected by the first. The die has no memory.
When you want the probability that two independent events both happen, multiply their individual probabilities. Say you roll a standard six-sided die twice and want to know the odds of getting a 3 on the first roll and a 3 on the second roll. The probability of rolling a 3 is 1/6. The probability of rolling a 3 again is also 1/6. Multiply them: 1/6 × 1/6 = 1/36, or about 2.8 percent.
This works for any number of independent events. If you flip a coin three times and want all three to be heads, the probability is 1/2 × 1/2 × 1/2 = 1/8, or 12.5 percent. The more independent events you chain together, the less likely the whole sequence becomes.
Dependent events: drawing cards without replacing them
A dependent event is one where the outcome of the first event changes the odds of the second. Drawing a card from a deck, not putting it back, and drawing again—now the deck has 51 cards instead of 52, so the odds have shifted.
Say you have a standard 52-card deck and you want the probability of drawing two hearts in a row without replacing the first card. The probability of drawing a heart first is 13/52 (there are 13 hearts in 52 cards). If you drew a heart, there are now 12 hearts left and 51 cards total. The probability of drawing a second heart is 12/51. Multiply them: 13/52 × 12/51 = 156/2652, which simplifies to 12/204, or about 5.9 percent.
The key difference: after the first draw, the total number of cards changed, and the number of hearts changed. That's what makes it dependent. Always recalculate the second probability based on what actually happened in the first event.
Using a table to organize outcomes when there are many possibilities
When you have multiple events with multiple outcomes each, a table keeps you from missing any combination. Say you flip a coin and roll a die at the same time. The coin has 2 outcomes (heads, tails). The die has 6 outcomes (1, 2, 3, 4, 5, 6). Together, there are 2 × 6 = 12 possible outcomes.
| Coin | Die = 1 | Die = 2 | Die = 3 | Die = 4 | Die = 5 | Die = 6 |
|---|---|---|---|---|---|---|
| Heads | H, 1 | H, 2 | H, 3 | H, 4 | H, 5 | H, 6 |
| Tails | T, 1 | T, 2 | T, 3 | T, 4 | T, 5 | T, 6 |
Now if you want the probability of heads and an even number, count the favorable outcomes: H, 2; H, 4; H, 6. That's 3 outcomes out of 12 total. So the probability is 3/12 = 1/4, or 25 percent. A table makes it straightforward to see every possibility and count correctly.
Real-world examples: weather, sports, and games
A weather forecast says there's a 40% chance of rain tomorrow. That means, based on historical data and current conditions, rain occurs in 40 out of 100 similar weather patterns. It doesn't mean it will rain for 40% of the day—it means the probability is 0.4.
In sports, if a basketball player makes 75% of their free throws, the probability they make the next one is 0.75 (assuming conditions are similar). If they take two free throws in a row, the probability they make both is 0.75 × 0.75 = 0.5625, or about 56%. If they make the first, the probability they make the second is still 0.75 (assuming independence).
In games like poker or roulette, probability tells you the odds of each hand or spin. A roulette wheel with 38 slots (in American roulette) has a probability of 1/38 for any single number. A deck of cards has 4 aces out of 52 cards, so the probability of drawing an ace is 4/52 = 1/13, or about 7.7%. These calculations help you understand what fair odds should be.
Frequently Asked Questions
What's the difference between probability and odds?
Probability is a fraction of favorable outcomes over total outcomes. Odds compare favorable outcomes to unfavorable outcomes. If the probability of an event is 1/4, the odds are 1 to 3 (one favorable, three unfavorable). Probability ranges from 0 to 1; odds can be any ratio. Most people use probability in everyday language, but odds are common in gambling and betting.
Can probability be greater than 1?
No. A probability of 1 means the event will definitely happen. A probability greater than 1 is impossible and means you made a math error. Check that your favorable outcomes don't exceed your total outcomes, and that you divided correctly.
How do I calculate probability when I don't know the total number of outcomes?
You need to count or estimate the total. For a die, it's 6. For a deck, it's 52. For real-world events like "will it rain," you use historical data—how many times did it rain under these conditions, divided by how many times these conditions occurred. Without knowing the total, you can't compute probability.
What does a probability of 0.5 mean?
A probability of 0.5 (or 50%) means the event is equally likely to happen or not happen. A fair coin flip, a fair die roll landing on odd or even, or a 50-50 guess all have probability 0.5. It's the midpoint between impossible (0) and certain (1).
If I flip a coin and get heads five times in a row, is tails more likely next?
No. Each flip is independent—the coin has no memory. The probability of tails on the next flip is still 0.5. This mistake is called the gambler's fallacy. Past results don't change the odds of independent events, even if a streak feels like it should.