Stat 265 - Unit 2 - Video Set 5 - Class Examples | University of Alberta

University:
University of Alberta
Course:
STAT 151 | Introduction to Applied Statistics I
Academic year:

2024
Views:

442

Pages:

14
Author:

Rain Haque

Stat 265 – Unit 2 – Video Set 5 – Class Examples 1) Multiplicative Law for Intersections : The probability of the intersection of two events A and B is P( A  B)  P( A) P( B | A)  P( B) P( A | B). Note : If A and B are independent then P( A  B)  P( A) P( B). Extension : P( A  B  C )  P( A) P( B | A) P(C | A  B). 2) Additive Law for Unions : The probability of the union of two events A and B is P( A  B)  P( A)  P( B)  P( A  B). Note : If A and B are disjoint then P( A  B)  P( A)  P( B). Extension : P( A  B  C )   P( A)  P( B)  P(C )    P( A  B)  P( A  C )  P ( B  C )    P ( A  B  C ) . Example A: Two countries, A and B, are competing in a series of X games. The country that wins the greatest number of games is declared the winner. If both countries win an equal number of games the series is a tie. The location of the next game will be determined by the winner of the previous game. That is, if country B wins game 1, game 2 will be played in country B. Suppose it is hypothesized that country A wins 55% of their games vs. country B when at home (in country A) and 30% of their games against country B when on the road (in country B). a) Suppose X = 2 games. What is the probability model for the result of the series (A wins, B wins, or it is a tie) if Game 1 is played in Country A? b) Suppose X = 3 games. If the series starts in country A, what is the probability that the winners will alternate? Example B: Consider the grid below. Suppose that starting at point A you can go one step up or one step to the right at each move. This is continued until the point labeled B is reached. Assume all paths from A to B are equally likely. a) What is the probability you pass through at least one of points C or D? b) Given that you passed through D, what is the probability that you also passed through C? Example C: A room full of n students all throw their OneCards in a hat. Then, they each draw one OneCard at random from the hat and hold on to it. If a student selects their own card it is considered a match. What is the probability at least one student has a match? a) Suppose n = 2: Let's name the students A, B, C, D, ...etc. If we have n = 2 students, there are 2 possible combinations of selections as described below: Combo A B # of matches 1 A B 2 2 B A 0 b) Suppose n = 3: If we have n = 3 students, there are 6 possible combinations of selections as described below: Combo 1 2 3 4 5 6 A A A B B C C B B C A C A B C # of matches C 3 B 1 C 1 A 0 B 0 A 1 c) Suppose n = ?: Example D: A group of 3 Americans, 4 Canadians and 8 Italians are to be broken up into five groups of three. What is the probability at least one group has a member from all three countries?