Exam 6: Probability

The following table shows the numbers of cars sold by a car dealer during the last 30 weeks. Number of cars sold Number of weeks 2 5 3 9 4 10 5 4 6 2 a. Define the random variable of interest to the dealer. b. List the simple events in the sample space. c. Assign probabilities to the simple events and show the probability distribution. d. What approach have you used in determining the probabilities in part (c)? e. What is the probability of selling no more than four cars in any given week?

An financial advisor tells you that in her estimation there is an 85% chance that a particular stock's price will increase over the next three weeks. a. Which approach was used to produce this figure? b. Interpret the 85% probability.

Suppose P(A) = 0.10, P(B) = 0.70, and P(B/A) = 0.80. a. Find P(A $\cap$ \ B). b. Find P(A $\cup$ B). c. Find P(A | B).

Bayes' theorem allows us to compute conditional probabilities from other forms of probability.

Conditional probability is the probability that an event will occur, given that another event will also occur.

An effective and simple method of applying the probability rules is the probability tree, wherein the events of an experiment are represented by lines.

If event A does not occur, then its complement $\overline { {A} }$ must occur.

Assume that A and B are independent events, with P(A) = 0.30 and P(B) = 0.50. The probability that both events will occur simultaneously is 0.80.

A table of joint probabilities is shown below. 0.15 0.25 0.20 0.10 0.15 0.15 Calculate P( $A _ { 3 }$ / $B _ { 1 }$ ).

A woman is expecting her second child. Her doctor has told her that she has a 50-50 chance of having another girl. If she has another girl, there is a 90% chance that she will be taller than the first. If she has a boy, however, there is only a 25% chance that he will be taller than the first child. Find the probability that the woman's second child will be taller than the first.

There are three approaches to determining the probability that an outcome will occur: the classical, relative frequency, and subjective approaches. Which is most appropriate in determining the probability of the following outcomes? a. A flipped coin will land on tails. b. The probability of your favourite team winning the finals. c. Five of the next 20 new cars sold in Adelaide will be imported cars.

If A and B are independent events, with P(A) = 0.20 and P(B) =0.60, then P(A | B) is: A. 0.20. B. 0.60. C. 0.40. D. 0.80.

An ice cream vendor sells three flavours: chocolate, strawberry and vanilla. 45% of the sales are chocolate, 30% are strawberry and the rest are vanilla. Sales are by the cone or the cup. The percentages of cones sales for chocolate, strawberry and vanilla are 75%, 60% and 40%, respectively. For a randomly selected sale, define the following events: A₁ = chocolate chosen. A₂ = strawberry chosen. A₃ = vanilla chosen. B = ice cream in a cone. $\bar { B }$ = ice cream in a cup. Use this information to answer the following question(s). -Find the probability that the ice cream was vanilla-flavoured, given that it was sold in a cup.

An experiment consists of three stages. There are two possible outcomes in the first stage, three possible outcomes in the second stage, and four possible outcomes in the third stage. Drawing a tree diagram for this experiment will show that the total number of outcomes is: A. 9. B. 24. C. 26. D. 18.

Of the last 400 customers entering a supermarket, 20 have purchased a mobile phone. If the classical approach for assigning probabilities is used, the probability that the next customer will purchase a mobile phone is: A. 0.80. B. 0.20. C. 0.05 D. 0.50.

An investment firm has classified its clients according to their gender and the composition of their investment portfolios (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The proportions of clients falling into the various categories are shown in the following table: Portfolio Composition Gender Bonds Stocks Balanced Male 0.18 0.20 0.25 Female 0.12 0.10 0.15 One client is selected at random, and two events A and B are defined as follows: A: The client selected is male. B: The client selected has a balanced portfolio. Find the following probabilities. a. P(A | B). b. P(B | A). c. P(A | $\bar { B }$ ). d. P( $\bar{A}$ | B).

Suppose P(A) = 0.1, P(B) = 0.5, and P(A $\cap$ B) = 0. a. Find P(A $\cup$ B). b. Are A and B independent events? Explain. c. Are A and B mutually exclusive events? Explain.

A useful graphical method of constructing the sample space for an experiment is:

An experiment consists of tossing three fair (unbiased) coins simultaneously. This experiment has eight possible outcomes.

An insurance company has collected the following data on the gender and marital status of 300 customers. Marital Status Gender Single Married Divorced Male 25 125 30 Female 50 50 20 Suppose that a customer is selected at random. Find the probability that the customer selected is: a. a married female. b. not single. c. married, if the customer is male. d. female or divorced. e. Are gender and marital status mutually exclusive? Explain using probabilities. f. Is marital status independent of gender? Explain using probabilities.