A sample space lists all outcomes of a random process, and an event is a subset of it. Probabilities follow rules: the complement rule, the addition rule \(P(A)+P(B)-P(A\text{ and }B)\), and conditional probability \(P(A|B)\). Events are independent when \(P(A\text{ and }B)=P(A)P(B)\), which means one event does not change the other’s probability.
Key facts:
- Standards S-CP.A.1 to S-CP.A.5, S-CP.B.6 and S-CP.B.7: events, independence, conditional probability and the addition rule.
- Complement rule: \(P(\text{not }A)=1-P(A)\).
- Addition rule: \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\); subtract the overlap once.
- Conditional probability: \(P(A|B)=\frac{P(A\text{ and }B)}{P(B)}\), the fraction of \(B\) that is also in \(A\).
- Test independence by checking \(P(A\text{ and }B)=P(A)\cdot P(B)\) or \(P(A|B)=P(A)\).
What are a sample space and an event?
The sample space is the set of all possible outcomes of a random process. An event is a subset of the sample space. When all outcomes are equally likely, \(P(E)=\frac{\text{outcomes in } E}{\text{outcomes in the sample space}}\). Rolling two dice has \(36\) outcomes, and \(6\) of them add to \(7\), so \(P(\text{sum is }7)=\frac{6}{36}=\frac{1}{6}\).
How do unions, intersections and complements work?
A and B (intersection): the outcomes that are in both events.
A or B (union): the outcomes in at least one of the events.
Not A (complement): every outcome outside \(A\), with \(P(\text{not }A)=1-P(A)\).
Suppose \(200\) students are surveyed. \(80\) are in band, \(100\) play a sport and \(30\) do both. The addition rule avoids counting the overlap twice: \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\). Here \(0.40+0.50-0.15=0.75\).
What is conditional probability?
The probability of \(A\) given \(B\) restricts attention to the outcomes in \(B\): \(P(A|B)=\frac{P(A\text{ and }B)}{P(B)}\). A two-way table shows this directly. Among the \(80\) band students, \(30\) play a sport, so \(P(\text{sports}\,|\,\text{band})=\frac{30}{80}=\frac{3}{8}\).
When are two events independent?
Events are independent if knowing one happened does not change the probability of the other. Test with \(P(A\text{ and }B)=P(A)\cdot P(B)\), or equivalently \(P(A|B)=P(A)\). Band and sports are not independent: \(0.40\cdot 0.50=0.20\), but the table gives \(0.15\).
| Rule | Formula |
|---|---|
| Complement | \(P(\text{not }A)=1-P(A)\) |
| Addition | \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\) |
| Multiplication | \(P(A\text{ and }B)=P(A)\cdot P(B|A)\) |
| Independence | \(P(A\text{ and }B)=P(A)\cdot P(B)\) |
How do you use a tree diagram?
- Draw one branch for each possible first result and write its probability.
- From each branch, draw the second results with their conditional probabilities.
- Multiply along a path to get the probability of that outcome.
- Add the paths that belong to the event.
A bag holds \(5\) red and \(3\) blue marbles, and two are drawn without replacement. The probability of two reds is \(\frac{5}{8}\cdot\frac{4}{7}=\frac{5}{14}\). The probability of two different colors is \(\frac{15}{56}+\frac{15}{56}=\frac{15}{28}\).
Probability: Sample Spaces, Independence and Conditional Probability: frequently asked questions
What is the difference between independent and mutually exclusive events?
Mutually exclusive events cannot happen together, so \(P(A\text{ and }B)=0\). Independent events do not affect each other, so \(P(A\text{ and }B)=P(A)P(B)\). Two events with positive probabilities cannot be both.
Why do you subtract P(A and B) in the addition rule?
Outcomes in both events are counted once in \(P(A)\) and again in \(P(B)\). Subtracting \(P(A\text{ and }B)\) removes the double count. If \(A\) and \(B\) are mutually exclusive, the term is zero.
How do you find a conditional probability from a two-way table?
Use only the row or column for the given event as the denominator. For \(P(\text{sports}\,|\,\text{band})\), divide the band-and-sports count \(30\) by the band total \(80\), which gives \(\frac{3}{8}\).
How do you check whether two events are independent?
Compare \(P(A\text{ and }B)\) with \(P(A)\cdot P(B)\). If the two numbers are equal, the events are independent. Equivalently, check whether \(P(A|B)\) equals \(P(A)\).
When do the probabilities on a tree diagram change?
They change when the first result affects the second, as when drawing without replacement. Then the second-level branches show conditional probabilities. With replacement, the second-level probabilities repeat the first level.
- Probability: Sample Spaces, Independence and Conditional Probability worksheet (24 problems)
- Probability: Sample Spaces, Independence and Conditional Probability quiz (10 questions)
- Probability: Sample Spaces, Independence and Conditional Probability test (45 minutes)
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