4.3: Some Elementary Probability Rules Flashcards
What is the rule of complements in probability?
The rule of complements states that the probability of an event not occurring is 1 minus the probability of the event occurring, expressed as P(A’) = 1 - P(A).
How do you interpret the complement of an event in a Venn diagram?
In a Venn diagram, the complement of an event A, denoted as A’, is the region that includes all sample space outcomes not in A. The probabilities of A and A’ add up to 1.
Define the intersection of two events.
The intersection of two events A and B, denoted as A ∩ B, is the set of outcomes in which both events occur simultaneously. P(A ∩ B) gives the probability of both events happening together.
How do you calculate probabilities using a contingency table?
A contingency table helps organize the frequencies of different outcomes.
To find the probability of an event, divide the number of favorable outcomes by the total number of outcomes.
For joint probabilities, divide the intersection count by the total.
Define the union of two events.
The union of two events A and B, denoted as A ∪ B, is the event that occurs if either A or B (or both) occur.
P(A ∪ B) is the probability that at least one of the events will occur.
What is the Addition Rule in probability?
The Addition Rule states that the probability of A or B occurring is the sum of the probabilities of each event minus the probability of their intersection:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
How do you calculate the probability of the union of events from a contingency table?
Add the probabilities of the individual events and subtract the probability of their intersection, as found in the contingency table, to avoid counting the intersection twice.
What does it mean when two events are mutually exclusive?
Two events A and B are mutually exclusive if they cannot occur at the same time, which means their intersection is empty: P(A ∩ B) = 0.
How are Venn diagrams used to represent the probability of events A, B, A ∩ B, and A ∪ B?
Venn diagrams visually represent events in probability. ‘A’ is the region exclusive to event A, ‘B’ to event B, ‘A ∩ B’ the overlap showing where both events occur, and ‘A ∪ B’ covers all areas of A and B combined.
What does the shaded area in a Venn diagram represent?
The shaded area in a Venn diagram represents the occurrence of the event(s).
For example, if the shaded area covers A ∪ B, it represents all outcomes included in either A, B, or both.
How do you calculate the probability of either event A or event B occurring using a Venn diagram?
To calculate P(A ∪ B), you add P(A) and P(B) and subtract P(A ∩ B), which is represented by the overlap in the Venn diagram.
What is the Addition Rule for two mutually exclusive events?
If events A and B are mutually exclusive, meaning they cannot occur together (no overlap in the Venn diagram), then P(A ∪ B) = P(A) + P(B).
Explain the concept of mutually exclusive events with an example.
Mutually exclusive events cannot happen at the same time.
For instance, when drawing a card, it cannot be a Jack and a Queen simultaneously; hence, the events ‘drawing a Jack’ and ‘drawing a Queen’ are mutually exclusive.
What is the probability of drawing a Jack or a Queen from a standard deck of playing cards?
The probability P(J ∪ Q) is calculated as P(J) + P(Q) because they are mutually exclusive events.
With 4 jacks and 4 queens in a 52-card deck, P(J ∪ Q) = 4/52 + 4/52 = 8/52 = 2/13.
How do you calculate the probability of drawing a Jack or a red card?
Since Jacks and red cards are not mutually exclusive, calculate P(J ∪ R) as P(J) + P(R) – P(J ∩ R).
With 4 jacks, 26 red cards, and 2 red jacks, P(J ∪ R) = 4/52 + 26/52 – 2/52 = 28/52 = 7/13.
