Probability of working employees
1. Store A, B, and C have 50, 75, and 100 employees, respectively. At store A 50% of the employees are women, at store B 60 % of the employees are women, and at store C 70% of the employees are women. Resignations are equally likely for all employees, regardless of gender. One employee resigns, and this employee is a woman. What is the probability that the woman worked at store C? at store B? at store A?
2. Two balls are chosen randomly and, therefore without replacement from an urn containing 8 white, 4 black and 2 orange balls.
a. List the possible outcomes, where outcomes indicate the colors 0f the two balls chosen. You may do this by indicating the color of the first ball chosen and the color of the second ball chosen, or you may just indicate the colors observed (in some order)
b. Assign probabilities to the outcomes you have included in your sample space in (a)
c. Suppose that we win $2 for each black ball selected and lose $1 for each white ball selected. Let Y denote our winnings of one play of this game. What are the possible values for Y and the associated probabilities for each of these values, i.e., what is the probability distribution for Y?
3. In a league championship series, games are played until one team has won 2 games, called a best 2 of 3 series. Suppose Team A wins each game with probability 0.6, Team B with probability 0.4. Suppose also that the outcomes of successive games are independent of one another. Determine
a. the outcomes that are possible
b. b. the probability that Team A wins the series
c. c. the probability that the series requires 2 games? 3 games?
4. In a league championship series, games are played until one team has won 2 games, called a best 2 of 3 series. Suppose Team A wins the first game with probability 0.6, and wins successive games with probability 0.8 if it won the previous game, with probability 0.5 if the lost the previous game.
a. How likely is it that Team A wins the second game of the series?
b. How likely is it that Team A wins the series?
c. How likely is it that the series lasts 2 games? 3 games? 4 games?
5. A box contains 6 marbles, two red, two green and two blue. Consider an experiment that consists of taking one marble from the box, then replacing it in the box and drawing a second marble from the box.
a. List the points in the sample space and assign probabilities to the points.
b. If Y is the random variable that indicates the number of different colors of marbles observed in the sample, find the probability mass function for Y
6. In the situation posed in Problem #5, suppose the second draw is made without replacing the first drawn marble
a. How do the sample points and/or probabilities assigned to them change?
b. What is the probability distribution for Y, defined as in Problem #5?
c. Thus, does it matter whether sampling is done with replacement or without replacement in this situation?
Please show all steps with answers. Thanks.