All Labs

Probability & Combinatorics Lab

Calculate union, intersection, and conditional probabilities with interactive Venn diagrams. Count permutations and combinations step by step. Explore Pascal's triangle and discover its connection to binomial coefficients.

Guided Experiment: Complement Rule and Addition Rule

How does P(A') relate to P(A)? How does P(A∪B) relate to P(A), P(B), and P(A∩B) for different types of events (independent, mutually exclusive, overlapping)?

Write your hypothesis in the Lab Report panel, then click Next.

U0.240A0.360B0.160A∩B0.240

Controls

Results

P(A)
0.6000
P(B)
0.4000
P(A∩B)
0.2400
P(A∪B)
0.7600
P(A|B)
0.6000
P(B|A)
0.4000
P(A')
0.4000
P(B')
0.6000
Independent
Step-by-Step
Events are independent, so P(A∩B)=P(A)×P(B)=0.6000×0.4000=0.2400\text{Events are independent, so } P(A \cap B) = P(A) \times P(B) = 0.6000 \times 0.4000 = 0.2400
P(A∪B)=P(A)+P(B)−P(A∩B)=0.6000+0.4000−0.2400=0.7600P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.6000 + 0.4000 - 0.2400 = 0.7600
P(A∣B)=P(A∩B)P(B)=0.24000.4000=0.6000P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2400}{0.4000} = 0.6000
P(B∣A)=P(A∩B)P(A)=0.24000.6000=0.4000P(B \mid A) = \frac{P(A \cap B)}{P(A)} = \frac{0.2400}{0.6000} = 0.4000
P(A′)=1−P(A)=1−0.6000=0.4000P(A') = 1 - P(A) = 1 - 0.6000 = 0.4000
P(B′)=1−P(B)=1−0.4000=0.6000P(B') = 1 - P(B) = 1 - 0.4000 = 0.6000
Independence check: P(A∩B)=0.2400≈P(A)×P(B)=0.2400  ✓\text{Independence check: } P(A \cap B) = 0.2400 \approx P(A) \times P(B) = 0.2400 \; \checkmark

Data Table

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Reference Guide

Probability Rules

The addition rule for two events uses inclusion-exclusion to avoid double-counting the overlap.

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

The complement rule says the probability of an event not happening is one minus the probability it does happen.

Conditional Probability

Conditional probability measures the likelihood of A given that B has already occurred.

P(A∣B)=P(A∩B)P(B)P(A | B) = \frac{P(A \cap B)}{P(B)}

If events are independent, knowing B gives no information about A, so P(A|B) = P(A).

Permutations & Combinations

Permutations count ordered arrangements. Combinations count unordered selections.

P(n,r)=n!(n−r)!,C(n,r)=n!r!(n−r)!P(n,r) = \frac{n!}{(n-r)!}, \quad C(n,r) = \frac{n!}{r!(n-r)!}

Use permutations when order matters (passwords, rankings). Use combinations when order does not matter (committees, card hands).

Pascal's Triangle

Each entry in Pascal's triangle equals the sum of the two entries above it, and also equals a binomial coefficient.

(nr)=(n−1r−1)+(n−1r)\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}

Row n sums to 2 to the n. The entries of row n give the coefficients of (a+b) to the n.