Recitation-4 (Probability Recall) Flashcards
(20 cards)
What is a sample space in probability, and how do you recognize it in a problem?
The sample space is the complete set of all possible outcomes of a random experiment. Notation: S. E.g., for rolling one die: S = {1,2,3,4,5,6}
How is an event different from the sample space?
An event is any subset of the sample space. It represents outcomes we’re interested in. Example: Event ‘sum is 7’ when rolling 2 dice: E = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
What do the operations ∪, ∩, and complement mean for events?
Union: A ∪ B → either A or B happens. Intersection: A ∩ B → both A and B happen. Complement: Aᶜ → event A does not happen.
What are the three basic properties (axioms) of a probability function?
Non-negativity: P(E) ≥ 0. Normalization: P(S) = 1. Additivity: If E₁ ∩ E₂ = ∅ → P(E₁ ∪ E₂) = P(E₁) + P(E₂)
What is a random variable, and how is it used in problems?
A random variable X maps outcomes (in S) to real values. It summarizes outcomes numerically. E.g., X = sum of two dice rolls → X(1,3) = 4
How do you calculate the probability of a random variable taking a specific value?
Add up the probabilities of all outcomes that map to that value. E.g., P(X = 4) = P((1,3)) + P((2,2)) + P((3,1))
What is the expected value of a discrete random variable?
E(X) = ∑ x · P(x). It gives the average value you’d expect over many trials.
What is the expected value in the continuous case?
E(X) = ∫ x · f(x) dx, where f(x) is the probability density function (pdf).
What are the formulas for variance and standard deviation?
Variance: σ² = E[(X - μ)²]. Standard deviation: σ = √Var(X)
What’s the formula for P(A ∪ B)?
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
What is the expanded formula for three events?
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C)
What is conditional probability and its formula?
P(A|B) = P(A ∩ B) / P(B). It’s the chance of A happening given B already happened.
If 90% of students pass, and 90% of those studied, what’s P(Pass ∩ Learn)?
P(Pass ∩ Learn) = P(Pass) × P(Learn | Pass) = 0.9 × 0.9 = 0.81
How can you compute the total probability of an event from joint probabilities?
Use: P(Learn) = P(Pass ∩ Learn) + P(Fail ∩ Learn)
How do you compute conditional probability in reverse?
Use Bayes-like logic: P(Pass | Learn) = P(Pass ∩ Learn) / P(Learn)
When are two events A and B independent?
If: P(A ∩ B) = P(A) × P(B). Also implies: P(A | B) = P(A)
What’s the key difference between independence and mutual exclusivity?
Independent: knowing one doesn’t affect the other. Mutually exclusive: both can’t happen at the same time.
What is the shortcut to calculate conditional probability from tables or examples?
Look for: P(A ∩ B) / P(B) → focus on joint over condition base
What’s a good memory hook for conditional probability?
Think: ‘Of the times B happens, how often does A also happen?’ That’s P(A|B)!
What is a common mistake when using conditional probability?
Reversing the condition: P(A|B) ≠ P(B|A). Always check which is known, and which is being asked.
