Equally likely means two or more outcomes have the same probability of happening. In Intro to Statistics, you use it when modeling fair experiments, like a fair die or a uniform distribution.
Equally likely means each outcome, or each value in a specified range, has the same probability in an Intro to Statistics problem. If a die is fair, each face is equally likely because no result is favored over another. If a random number is chosen from a uniform distribution, every interval of the same length has the same chance of being selected.
This idea shows up most often when you are building a probability model from scratch. You first decide what the outcomes are, then ask whether the situation treats them all the same. If it does, you can assign equal probabilities and check that they add up to 1. For a fair six-sided die, each outcome gets probability 1/6, because there are six equally likely outcomes and 6 times 1/6 = 1.
The phrase can also describe continuous random variables. In a uniform distribution on [a, b], not every single value is a separate outcome you can count the way you do with a die. Instead, equal-length intervals are equally likely. So the interval from 2 to 4 has the same probability as the interval from 7 to 9, as long as both intervals have the same length and stay inside the range.
A common trap is thinking equally likely means you will see each outcome the same number of times in a small sample. That is not what the term says. It describes the theoretical model, not a guarantee about one short run of trials. You might roll a fair die ten times and get four 6s, but that does not mean the die was unfair.
In probability, equally likely is a setup idea. Once you know the outcomes are equally likely, you can count favorable outcomes, divide by the total number of outcomes, and move into probability calculations with a clean model.
Equally likely is one of the first checks you make before solving probability problems in Intro to Statistics. If a situation is equally likely, you can use simple counting or a uniform distribution model instead of guessing weights for different outcomes.
That matters because a lot of early stats work depends on setting up the sample space correctly. If you treat an unfair situation as equally likely, your probabilities will be wrong from the start. If you recognize that a fair die, a coin, or a random choice from a uniform interval has equal chances built in, the rest of the calculation becomes much cleaner.
It also connects directly to how statistics describes randomness. Equal likelihood is about the model, while long-term relative frequency is about what happens after many trials. A good stats class keeps those ideas separate so you do not confuse theoretical probability with results from one short experiment.
You will also see this idea when interpreting graphs and distribution shapes. Uniform distributions are flat because every value or equal-length interval has the same chance. That makes equally likely a bridge between probability language and visual interpretation.
Keep studying Intro to Statistics Unit 3
Visual cheatsheet
view galleryProbability
Equally likely is one way probability gets calculated. When all outcomes have the same chance, you can use favorable outcomes divided by total outcomes. If the outcomes are not equally likely, you need a different model, so spotting this condition first saves you from using the wrong formula.
Uniform Distribution
A uniform distribution is the continuous version of equally likely behavior. Instead of each discrete outcome getting the same probability, every interval of the same length inside the range gets the same probability. That is why the graph is flat across the interval.
Continuous Random Variable
For a continuous random variable, you do not assign probability to one exact value the way you do with a die face. Equally likely usually means equal-length intervals have equal probability. This changes how you think about the sample space and why single points have probability 0.
long-term relative frequency
Equally likely describes the theoretical model, while long-term relative frequency describes what happens after many repetitions. If outcomes are equally likely, the relative frequencies should settle near the same proportions over time, but a short experiment can look uneven.
A quiz or problem set question will usually ask you to decide whether a situation has equally likely outcomes before you calculate probability. You might be given a spinner, die, or random interval and asked to identify the sample space, check whether the outcomes are fair, or explain why a uniform model fits. The move is simple: name the outcomes, decide whether each one has the same chance, then use that fact to find probabilities.
You may also see a question that tests the difference between theoretical probability and observed results. If a simulation or class experiment gives uneven counts, you should not automatically conclude the outcomes were not equally likely. Short-run variation is normal, so the correct answer usually refers back to the model, not just the sample data.
Equally likely is about the probability built into the model, while long-term relative frequency is about the pattern you expect after many trials. A fair die has equally likely faces even if one class set of rolls comes out lopsided. The model stays the same, but observed frequency can bounce around in the short run.
Equally likely means the outcomes have the same probability in the probability model, not that they will appear evenly in a small sample.
For a fair die, each face is equally likely, so each outcome has probability 1/6.
In a uniform distribution, equal-length intervals have equal probability across the range.
The probabilities of all equally likely outcomes must add up to 1.
This term helps you decide whether to use simple counting or a uniform model before you calculate anything.
Equally likely means the outcomes in a probability model all have the same chance of happening. In Intro to Statistics, you see this with fair dice, fair coins, and uniform distributions over an interval. Once you know outcomes are equally likely, probability calculations are usually much simpler.
No. Equally likely is a statement about the probability model, while long-term relative frequency describes what happens after many trials. A fair process can still give uneven results in a short run, so one experiment does not prove the outcomes were not equally likely.
Count the favorable outcomes and divide by the total number of equally likely outcomes. For example, on a fair die, rolling an even number gives 3 favorable outcomes out of 6 total outcomes, so the probability is 3/6 or 1/2.
In a uniform distribution, every interval with the same length has the same probability. That does not mean each exact value is counted separately the way discrete outcomes are. It means the probability is spread evenly across the whole interval.