chapter4 Flashcards
(18 cards)
What is an experiment in statistics?
An experiment is any process that generates a set of data.
Example: Tossing a coin (possible outcomes: Head or Tail).
Define outcomes and sample space.
Outcomes: Possible results of an experiment.
Sample Space (S): The collection of all possible outcomes.
Example: Tossing 2 coins → S = {HH, HT, TH, TT}.
What is an event?
An event is a subset of a sample space.
Types of Events:
Simple Event: Contains one sample point.
Compound Event: Contains two or more sample points.
Example: S = {1, 2, 3, 4, 5, 6}, Event A = rolling an even number = {2, 4, 6}.
What is the difference between mutually exclusive and independent events?
Mutually Exclusive: Events cannot happen at the same time.
Independent: One event does not affect the probability of the other.
Example: Rolling an even number (A) and an odd number (B) with a die; Tossing a coin and rolling a die.
What is the union of two events?
The union of events A and B (A ∪ B) is the event that either A, B, or both occur.
Formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Example: A = {2, 4, 6}, B = {1, 3, 5}. A ∪ B = {1, 2, 3, 4, 5, 6}.
What is the intersection of two events?
The intersection of events A and B (A ∩ B) is the event where both A and B occur.
Formula: P(A ∩ B) = P(A) × P(B|A).
Example: Event A = Engineering students, Event B = Female students. A ∩ B = Female engineering students.
Explain complementary events.
The complement of event A (denoted A’) is the event that A does not occur.
Example: Sample Space: {Heads, Tails}. Event A = {Heads}. Complement of A = {Tails}.
What is classical probability?
Classical probability assumes all outcomes are equally likely.
Formula: P(A) = (Number of outcomes favorable to A) / (Total outcomes).
Example: Rolling a die, P(rolling a 4) = 1/6.
What is the additive rule of probability?
Used to find the probability of the union of events.
Formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Example: A = rolling an even number = {2, 4, 6}. B = rolling a number < 4 = {1, 2, 3}. P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Define conditional probability.
The probability of event A occurring given that event B has occurred.
Formula: P(A|B) = P(A ∩ B) / P(B).
Example: What is the probability of drawing an ace given the card is black? P(Ace|Black) = (2/52) / (26/52) = 2/26 = 1/13.
What are factorials, combinations, and permutations?
Factorial (n!): Product of all positive integers up to n.
Combination: Selection of items where order doesn’t matter.
Permutation: Selection of items where order matters.
Example: 4! = 4 × 3 × 2 × 1 = 24; C(n, r) = n! / [r!(n - r)!]; P(n, r) = n! / (n - r)!
How do you check if two events are independent?
Two events A and B are independent if: P(A | B) = P(A) or P(B | A) = P(B).
Example: P(A) = 0.3, P(A ∩ B) = 0.09. Check: P(A ∩ B) = P(A) × P(B). If true, they are independent.
What is the multiplicative rule of probability?
Used to find the probability of the intersection of two events.
Formula: P(A ∩ B) = P(A) × P(B|A).
For Independent Events: P(A ∩ B) = P(A) × P(B).
What are disjoint events?
Disjoint events (mutually exclusive) cannot occur at the same time.
Example: When rolling a die: Event A = rolling an even number = {2, 4, 6}. Event B = rolling an odd number = {1, 3, 5}. A and B are disjoint since they cannot overlap.
Define random variable.
A random variable assigns numerical values to outcomes of a random experiment.
Types: Discrete: Takes specific values (e.g., number of heads). Continuous: Takes any value within a range (e.g., height).
What is a probability distribution?
A probability distribution shows all possible values of a random variable and their probabilities.
Example: Rolling a die: X = Outcome, P(X) = 1/6 for each outcome {1, 2, 3, 4, 5, 6}.
What are the properties of probability distributions?
The probability of each outcome is between 0 and 1. The sum of all probabilities is 1.
What is the difference between population and sample?
Population: The entire group being studied.
