Combination With Repetitions (Counting Rules + Video 4 & 5) Class Notes

University:
McMaster University
Course:
STATS 1LL3 | Introduction to Probability and Statistics
Academic year:

2023
Views:

204

Pages:

3
Author:

Sabrina S.

"We have n types of objects, with at least available of each type. The number of selections of r objects, allowing repetitions is given by ()" "the proof of this result is included as an exercise at end of chapter" Counting Rules: Rule 1: "the number of distinguishable arrangements of n distinct objects, not allowing repetitions is n!" Rule 2: "The number of ways of ordering r objects chosen from n distinct objects, not allowing repetitions is (n)=(nr)!" Rule 3: "The number of ways of choosing a set of r objects from n distinct objects, not allowing repetitions is ()()()" Rule 4: "The number of permutations of r objects, chosen from n distinct objects allowing repetitions is " Video: "Video 4. Permutations and combinations" Pr(A): "(number of equally likely elementary events in A) / (number of equally likely elementary events in S)" What is a permutation? "An arrangement of objects where order does matter." What is a combination? "A selection of objects where order does not matter." What is conditional probability? "It is a method of updating our knowledge of the probability of an event when we are provided with new information concerning another event and this other event might or might not occur." The probability of event A occurring if event B has occurred: (Pr(A|B) = \frac{Pr(A \cap B)}{Pr(B)}) If A and B are unrelated or independent: (Pr(A|B) = Pr(A)) (Pr(A \cap B) = Pr(A) * Pr(B)) When two events are unrelated or independent, THEN the probability of intersection is equal to the product of their individual probabilities. Unit 4: Conditional probability Conditional probability: "Provide a method for updating or revising probabilities in the light of new information." "In statistical jargon, we say a weather forecast is conditional on the info available up to that point in time." Let A and B be two events in a sample space S. Then the conditional probability of the event B given the event A has occurred, denoted by (Pr(B|A)), is: (Pr(B|A) = \frac{Pr(B \cap A)}{Pr(A)}) "provided that (Pr(A) \neq 0). (Pr(B|A)) is read 'the probability of B given A.'" "The conditional probability (Pr(B|A)) may be thought of as a reassessment of the probability of B given the information that some other event A occurred." Example: (Pr(\text{King of clubs}|\text{Clubs}) = \frac{Pr(\text{King of clubs} \cap \text{Clubs})}{Pr(\text{Clubs})}) "The event 'King of clubs' is a subset of the events 'clubs,' so the intersection of these two events is the event 'King of clubs.'" (Pr(\text{Clubs}) = \frac{13}{52} = \frac{1}{4}) (Pr(\text{King of clubs}|\text{Clubs}) = \frac{1/52}{1/4} = \frac{1}{13}) Example: "Suppose A and B are two events in a sample space, and that (Pr(A) = 0.6), (Pr(B) = 0.2), and (Pr(A|B) = 0.5)." (Pr(A \cap B) = Pr(A|B) \cdot Pr(B) = 0.5 \cdot 0.2 = 0.1) (Pr(A \cup B) = Pr(A) + Pr(B) - Pr(A \cap B) = 0.6 + 0.2 - 0.1 = 0.7) (Pr(\overline{A} \cap B) = Pr(B) - Pr(A \cap B) = 0.2 - 0.1 = 0.1) (Pr(B|\overline{A}) = \frac{Pr(B \cap \overline{A})}{Pr(\overline{A})})