Module 4 Flashcards
What are exhaustive events in probability?
Exhaustive events include all possible outcomes of an experiment.
They cover the entire sample space.
What are mutually exclusive events in probability?
Mutually exclusive events do not share any common outcomes.
The occurrence of one event precludes the occurrence of the other.
Are the following events mutually exclusive?
Grades of A, B, C
Yes, because a student cannot receive both an A and a B simultaneously.
However, they are not exhaustive, as other grades (D, F) are possible.
What is the union (A ∪ B) of two events?
The union of two events includes all outcomes in either A or B, without double-counting.
Example: A ∪ B = {gold, silver, bronze, no medal}.
What is the intersection (A ∩ B) of two events?
The intersection of two events includes only the common outcomes in both events.
Example: A ∩ B = {silver, bronze}.
What is the complement (Bc) of an event?
The complement of an event includes all outcomes not in the event.
Example: Bc = {gold}, if B = {silver, bronze, no medal}.
How do you calculate P(A) for a snowboarder’s medal probabilities?
P(A) = P({gold}) + P({silver}) + P({bronze}) = 0.10 + 0.15 + 0.20 = 0.45.
What is empirical probability?
Empirical probability is based on observing data or the relative frequency of an event occurring.
It’s calculated by repeating an experiment many times.
How would you calculate the probability that an individual is between 50 and 60 years old, based on a frequency distribution?
Use the relative frequency formula:
P(50 ≤ age < 60) = (Number of individuals aged 50–60) / (Total number of observations).
Given sample space S = {gold, silver, bronze, no medal}, how do you calculate P(B ∪ C)?
P(B ∪ C) = P({silver}) + P({bronze}) + P({no medal}) = 0.15 + 0.20 + 0.55 = 0.90.
What is the rule of addition for calculating the probability of at least one event happening?
The probability of event A or event B happening is:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) = 0.75 + 0.55 - 0.40 = 0.90.
How do you calculate the probability that no event occurs?
The probability that neither A nor B happens is the complement:
P(not A ∩ not B) = 1 - P(A ∪ B).
1 - 0.90 = 0.10.
What is conditional probability?
The probability of event A occurring given that event B has already occurred is:
P(A | B) = P(A ∩ B) / P(B).
How do you determine if two events are independent?
Two events A and B are independent if:
P(A ∩ B) = P(A) * P(B).
Example: If the probability of your desktop crashing is 0.02, and the probability of your laptop crashing is 0.06, and the probability of both crashing is 0.0012, check if they are independent:
P(A) * P(B) = 0.02 * 0.06 = 0.0012 → They are independent.
How do you calculate the probability of both events A and B occurring?
The probability of both events occurring is:
P(A ∩ B) = P(A) * P(B) for independent events.
If they are not independent, use the rule of addition:
P(A ∩ B) = P(A) + P(B) - P(A ∪ B).
What are sample spaces and events?
Sample Space (S): All possible outcomes of an experiment.
Event: A subset of the sample space (e.g., A, B).
What is the difference between exhaustive and mutually exclusive events?
Exhaustive Events: Include all possible outcomes in the sample space.
Mutually Exclusive Events: Cannot occur simultaneously (no common outcomes).
What is the addition rule?
The probability of event A or event B occurring:
P(A∪B)=P(A)+P(B)−P(A∩B)
Example:
P(A) = 0.75, P(B) = 0.55, P(A ∩ B) =
P(A∪B)=0.75+0.55−0.40=0.90
What is the complement rule?
The probability of an event not occurring:
P(notA)=1−P(A)
Example:
If P(A) = 0.90, then P(not A) = 1 - 0.90 = 0.10.
What is conditional probability?
The probability of event A given that event B has occurred:
P(A∣B)= P(B)/P(A∩B)
Example:
P(A ∩ B) = 0.16, P(B) = 0.25
P(A∣B)= 0.25/0.16 =0.64
Joint probability
Joint Probability (P(A ∩ B)): The probability that both events A and B occur together.
Union probability
Union Probability (P(A ∪ B)): The probability that either event A or event B (or both) occurs.
P(A∪B)=P(A)+P(B)−P(A∩B)
What is a contingency table, and how is it used in statistics?
A contingency table displays the frequency distribution of two or more categorical variables. It helps calculate joint, marginal, and conditional probabilities by organizing data into rows and columns.
What is the difference between joint, marginal, and conditional probability?
Joint Probability: Probability of two events occurring together (e.g., both being male and promoted).
Marginal Probability: Probability of a single event (e.g., probability of being promoted).
Conditional Probability: Probability of one event given that another has already occurred (e.g., probability of being promoted given gender).
How do you calculate the probability of multiple independent events occurring?
Multiply the number of possible outcomes for each event.
Example: Rolling 3 coins:
2×2×2=8 outcomes.
What is the formula for calculating factorials?
n!=n×(n−1)×(n−2)×⋯×1
Example: 4!=4×3×2×1=24
What is the difference between combinations and permutations?
Combinations: Order doesn’t matter
Permutations: Order matters
How do you calculate conditional probability?
What is the probability that a customer prefers chocolate frozen yogurt, given they are at least 25 years old?
Formula: P(C∣T)=P(C∩T)/P(T)
Result: P(C∣T) = 15/75 = 0.20
How do you calculate the probability of an event?
P(A) = number of favorable outcomes/total number of outcomes
What are the counting rules for calculating possible outcomes?
Multistep: Multiply the number of outcomes for each step.
Factorial: Use n! to calculate the number of ways to arrange objects.
Combination: Used when order doesn’t matter, C(n,k).
Permutation: Used when order matters, P(n,k).
How do you calculate the probability of multiple events?
Multiply the individual probabilities for independent events.
