Exam 6: Probability

A table of joint probabilities is shown below. 0.15 0.25 0.20 0.10 0.15 0.15 Calculate P( $A _ { 1 }$ | $B _ { 2 }$ ).

If A and B are independent events, with P(A) = 0.30 and P(B) = 0.50, then P(B | A) is 0.10.

Bayes' Law is a method of revising probabilities after another event has occurred.

Find the probability that the ice cream was sold on a cone.

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

Find the probability that the ice cream was vanilla-flavoured, given that it was sold on a cone.

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.

Is it possible to have two events for which $P ( A ) = 0.40 , P ( B ) = 0.50 , \text { and } P ( A \cup B ) = 0.20 ?$ Explain.

If a coin is tossed three times, and a statistician predicts that the probability of obtaining three heads in a row is 0.125, which of the following assumptions is irrelevant to his prediction?

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. Express each of the following events in words. a. $A \cup B$ b. $A \cap B$ c. $A \cap \bar { B }$ d. $\bar { A } \cup \bar { B }$

A table of joint probabilities is shown below. 0.15 0.25 0.20 0.10 0.15 0.15 Calculate P( $A _ { 1 }$ or $B _ { 2 }$ ).

Suppose A and B are two mutually exclusive events for which P(A) = 0.25 and P(B) = 0.60. a. Find $P ( A \nrightarrow B )$ b. Find $P ( A \cup B )$ c. Find $P ( A \mid B )$ d. Are A and B independent events? Explain using probabilities.

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?

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.

Probability refers to a number between 0 and 1 (inclusive), which expresses the chance that an event will occur.

Suppose P( $\bar { A }$ ) = 0.10, P( $\bar { B }$ | A) = 0.40, and P( $\bar { B }$ | $\bar { A }$ ) = 0.50. a. Find P(A). b. $\text { Find } P ( \bar { B } \cap A )$ c. Find $P ( \bar { B } \cap \bar { A } )$

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

Suppose P(A) = 0.40, P(B) = 0.50, and P(A  B) = 0.70. a. Find P(A  B). b. Find P(B | A). c. Are A and B independent events? Explain using probabilities.

At the beginning of each year, an investment newsletter predicts whether or not the stock market will rise over the coming year. Historical evidence reveals that there is a 75% chance that the stock market will rise in any given year. The newsletter has predicted a rise for 80% of the years when the market actually rose, and has predicted a rise for 40% of the years when the market fell. Find the probability that the newsletter's prediction for next year will be correct.

A table of joint probabilities is shown below. 0.15 0.25 0.20 0.10 0.15 0.15 Calculate P( $B _ { 2 }$ / $A _ { 1 }$ ).