Chapter 15: Probability Rules!
Independence will continue to be one of the most critical issues in the course.
Key Vocabulary:
trial- a single attempt or realization of a random phenomenon.
outcome- the result of a trial. can be either measured, observed, or reported for ann individual instance of that trial.
event- any set or collection of outcomes.
sample space- the collection of all possible outcomes, denoted as S
disjoint- shares no common outcome (M&M example in class where if you pick one M&M, it cant be red and orange)
mutually exclusive- two events that cannot occur at the same time with no common outcomes (similar to being disjoint)
General Addition Rule - To find the probability of events that are not disjoint (or in both circles in a Venn diagram), we add the probabilities of two events and then subtract our the probability that occurs in both events. P(A or B) = P(A) + P(B) - P(A and B)
independent- that the outcome of one trial does not influence the outcome of another
conditional probability- a probability that takes into account a given condition.
tree diagram- A display of conditional events or probablities that is helpful in thinking through conditioning
1. If events A and B are disjoint, then. If events A and B are NOT disjoint, explain why this formula does not work.
If Events A and B are NOT disjoint (meaning one can happen along with the other or there is something in the intersection of the venn diagram displaying events A and B) then when you are finding probabilities (by adding them) you will be counting that probability in the center of the venn diagram 2 times (which is bad).
(This is accounted for by subtracting that probability in the middle.)
2. In general (whether events are disjoint or not), what is the formula for finding ?
P(A or B) = P(A) + P(B) - P(A and B)
3. Explain the difference between the Addition Rule for disjoint events and the General Addition Rule.
The Addition Rule states that for two disjoint events, the probability that one or the other occurs is the sum of the probabilities of the two events. (Book example on p. 280 about the probability that a randomly selected student is either a sophmore (A) or a junior (B) therefore P(A or B) = P(A)+P(B) provided that A and B are disjoint.)
The General Addition Rule does not require disjoint events. The probabliities of the two events is added, then the probability of their intersection is subtracted. (Book example on p. 291 about the probability of randomly selecting a bill with an odd-numbered value (A) or a building on the reverse (B). $5 bill is inclueded in both sets therefore P(A or B)=P(A)+P(B)-P(A and B).)
4. What is meant by joint probability?
Probabilty of two different events that are related
5. What is meant by conditional probability?
A probability that takes into account a given condition is called a conditional probability. It is written P(BlA) and pronounced "the probability of B given A"
6. State the formula for finding conditional probability.
P(B/A) = P(A and B) / P(A)
7. Is the probability of “A given B” the same as the probability of “B given A?” Explain.
Yes, because due to the General Multiplication Rule, it doesn't matter which set is A and which set is B. You hvae to multiply the probability of A and the probability of B to find the outcome. The equation can be flipped around because the order in multiplication problems does not matter, unlike in subtraction and division problems.
8. In general (whether events are independent or not), what is the formula for finding ?
P(A and B) = P(A)x P(B/A)
9. Explain the difference between the Multiplication Rule for independent events and the General Multiplication Rule.
The Multiplication Rule for independent events states ethat if A and B are independent events, then the probablitlity of A and B is
P(A and B) = P(A) x P(B)
The General Multiplication Rule states that for any two events, A and B, the probablitlity of A and B is
P(A and B)=P(A) x P(BlA)
10. State the formula used to determine whether or not two events are independent.
Events A and B are independent whenever:
P(BlA)=P(B).
*REMEMBER DISJOINT EVENTS CANNOT BE INDEPEDENT*
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