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Probability measures how likely an event is: it ranges from 0 (impossible) to 1 (certain). To calculate it, first identify the outcomes that count, then decide whether the question asks for “and,” “or,” or a probability given extra information. Those distinctions determine which rule to use.
What is probability?
A probability assigns a value between 0 and 1 to an event. A value of 0 means the event cannot occur; 1 means it must occur. The probabilities of all outcomes in a complete sample space add up to 1.
For equally likely outcomes, divide the number of favorable outcomes by the total number of possible outcomes:
P(A) = number of outcomes in A ÷ total number of outcomes
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For example, with a fair six-sided die, the probability of rolling an even number is 3/6, or 1/2, because three of the six equally likely faces are even. This favorable-over-total shortcut only applies when the outcomes being counted are equally likely.
The complement of an event A, written Ac, is the event that A does not happen. Its probability is P(Ac) = 1 − P(A). This is useful when counting all the ways an event can fail is easier than counting all the ways it can happen. These foundational rules are summarized in University of Chicago courseware.
How do I calculate conditional probability?
Conditional probability is the probability of event A given that event B is known to have occurred. It is written P(A|B) and read “the probability of A given B.” The condition narrows the relevant sample space to outcomes where B happened.
When P(B) is not zero, calculate it as:
P(A|B) = P(A ∩ B) ÷ P(B)
Here, A ∩ B means that both A and B occur. The denominator, P(B), limits the calculation to cases in which B occurred; the numerator counts the cases in which both B and A occurred. The definition and its use appear in OpenStax’s introductory statistics text and MIT’s probability and statistics course materials.
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Repair common Windows errors and clear accumulated junk for a smoother, more stable PC - no reinstall needed.Free scan · no reinstallExample: conditioning on a coin toss
For three tosses of a fair coin, the probability of getting heads on all three is 1/8. If you are told the first toss was heads, only four equally likely outcomes remain for the full sequence: HHH, HHT, HTH, and HTT. Just one has three heads, so the conditional probability is 1/4. The new information changes the sample space and therefore the answer.
Example: drawing cards without replacement
In a standard 52-card deck, if the first card drawn is a spade and is not returned, 51 cards remain, including 12 spades. The probability that the second card is a spade given that the first was a spade is therefore 12/51. The first draw changes what is available on the second draw.
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What is the difference between independent and mutually exclusive events?
These terms describe different relationships between events. Independent events do not change each other’s probabilities; mutually exclusive events cannot happen together.
| Relationship | Meaning | Mathematical test |
|---|---|---|
| Independent | Knowing B occurred does not change the probability of A. | P(A|B) = P(A), or equivalently P(A ∩ B) = P(A)P(B) |
| Mutually exclusive | A and B cannot both occur. | P(A ∩ B) = 0 |
Two events with nonzero probabilities cannot be both independent and mutually exclusive: if they are mutually exclusive, their intersection is 0, while independence would make that intersection P(A)P(B), which is greater than 0. The exception is when at least one event has probability zero.
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Compare the probability of A before and after learning that B occurred. If P(A|B) equals P(A), the events are independent. Alternatively, compare P(A ∩ B) with P(A)P(B). If the values match, they are independent. If they do not, they are dependent.
When do I use the addition or multiplication rule?
Use the addition rule for an “A or B” question and the multiplication rule for an “A and B” question. The rules account for overlap and dependence rather than assuming events are mutually exclusive or independent.
| Question | Rule | What it accounts for |
|---|---|---|
| A or B (at least one occurs) | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | Subtracts the overlap, which would otherwise be counted twice. |
| A and B (both occur) | P(A ∩ B) = P(A|B)P(B) | Uses the probability of A under the condition that B occurred. |
For mutually exclusive events, P(A ∩ B) is zero, so the addition rule simplifies to P(A ∪ B) = P(A) + P(B). For independent events, P(A|B) equals P(A), so the multiplication rule simplifies to P(A ∩ B) = P(A)P(B). Do not use these simplified versions unless the relevant relationship is established.
Worked example: overlap matters
In an instructional example from OpenStax, suppose P(A) = 0.65, P(B) = 0.65, and P(B|A) = 0.90. The probability that both occur is P(A ∩ B) = P(B|A)P(A) = 0.90 × 0.65 = 0.585. The probability that at least one occurs is then 0.65 + 0.65 − 0.585 = 0.715.
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These events are dependent because 0.585 is not equal to 0.65 × 0.65 = 0.4225. They are not mutually exclusive because their intersection is not zero. The figures are instructional values, not real-world population estimates.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How do I know when to use Bayes’ theorem?
Use Bayes’ theorem when you know the probability of evidence given a cause, but need the probability of the cause given the evidence. It reverses the direction of a conditional probability:
P(A|B) = P(B|A)P(A) ÷ P(B)
In this form, P(A) is the prior probability of A before considering B; P(B|A) is the likelihood of observing B if A is true; and P(B) is the overall probability of observing B. Bayes’ theorem updates a prior belief using evidence, as described by University of Chicago courseware.
The key caution is that P(A|B) and P(B|A) are not interchangeable. Evidence can be common among people who have a condition, yet the condition can still be uncommon among people with that evidence if the condition’s starting prevalence is low. Ignoring that prior probability is the base-rate fallacy; MIT includes recognizing this error among its course learning goals.
Which probability method should I use?
Translate the wording into events before choosing a formula. Then check what is known about their relationship and whether a condition changes the relevant sample space.
- “A or B”: use the addition rule, subtracting any overlap. If the events are mutually exclusive, the overlap is zero.
- “A and B”: use the multiplication rule. Use P(A)P(B) only when independence is established; otherwise use a conditional probability.
- “A given B” or “if B has occurred”: use conditional probability, restricting the sample space to B.
- Known evidence, unknown cause: use Bayes’ theorem to reverse the conditional direction and include the prior probability.
A formula is often enough for one or two events. For a sequence of stages, a tree diagram can show conditional probabilities at each branch and the joint probability along a path. Tables can help organize joint, marginal, and conditional probabilities. Pearson’s introductory statistics material describes tree diagrams as a way to visualize these relationships, while MIT’s course materials recommend trees and tables for organizing conditional-probability calculations.
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