Exam 6: Probability

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

Jim and John go to a coffee shop during their lunch break and toss a coin to see who will pay. The probability that John will pay three days in a row is 0.125.

Which of the following is not an approach to assigning probabilities?

A pharmaceutical firm has discovered a new diagnostic test for a certain disease that has infected 1% of the population. The firm has announced that 95% of those infected will show a positive test result, while 98% of those not infected will show a negative test result. What proportion of test results are correct?

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

According to an old song lyric, 'love and marriage go together like a horse and carriage'. Let love be event A and marriage be event B. Events A and B cannot be mutually exclusive.

The relative frequency approach to probability depends on the law of large numbers.

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 chocolate-flavoured, given that it was sold in a cup.

The relative frequency approach is not useful in interpreting probability statements such as those heard from weather forecasters or scientists.

When it is not reasonable to use the classical approach to assigning probabilities to the outcomes of an experiment, and there is no history of the outcomes, we have no alternative but to employ the subjective approach.

The probability of event A and event B occurring must be equal to 1.

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. Are A and $\bar { B }$ mutually exclusive events? Explain.

A table of joint probabilities is shown below. 0.15 0.25 0.20 0.10 0.15 0.15 Are the events A and B independent? Explain.

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 probabilities in words. a. P(A | B). b. P(B | A). c. P(A | $\bar { B }$ ). d. P( $\bar{A}$ | 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 $B _ { 2 }$ ).

A statistics professor classifies his students according to their gender and the number of hours of paid work they do a week. The following table gives the proportions of students falling into the various categories. One student is selected at random. Paid Work (hours/week) Gender 0 1-8 9-16 Over 16 Male 0.06 0.20 0.15 0.06 Female 0.12 0.18 0.20 0.03 a. If the student selected is female, what is the probability that he works between 1 and 8 hours a week? b. If the selected student works more than 16 hours a week, what is the probability that the student is male? c. What is the probability that the student selected is female or does do any paid work or both? d. Is gender independent of the number of hours of paid work done a week? Explain using probabilities.

Suppose P(A) = 0.50, P(B) = 0.30, and P(A or B) = 0.80. a. Find P(A $\cap$ B). b. Find P(B | A). c. Are A and B mutually exclusive events? Explain using probabilities.

Two events A and B are said to be independent if P(A $\cap$ \B) = P(A) + P(B).

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. P(B). c. P( $\bar{A}$ ).

If A and B are mutually exclusive events, with P(A) = 0.20 and P(B) = 0.30, then P(A $\cap$ \ B) is: A. 0.50. B. 0.10. C. 0.00. D. 0.06.