91Ó°ÊÓ

You draw two cards from a standard deck of 52 cards without replacing the first one before drawing the second. (a) Are the outcomes on the two cards independent? Why? (b) Find \(P(3\) on 1 st card and 10 on 2 nd). (c) Find \(P(10\) on 1 st card and 3 on 2 nd \()\). (d) Find the probability of drawing a 10 and a 3 in either order.

You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck. (a) Are the outcomes on the two cards independent? Why? (b) Find \(P(\) ace on 1 st card and king on 2 nd). (c) Find \(P(\) king on 1 st card and ace on 2 nd ). (d) Find the probability of drawing an ace and a king in either order.

John runs a computer software store. Yesterday he counted 127 people who walked by his store, 58 of whom came into the store. Of the 58 , only 25 bought something in the store. (a) Estimate the probability that a person who walks by the store will enter the store. (b) Estimate the probability that a person who walks into the store will buy something. (c) Estimate the probability that a person who walks by the store will come in and buy something. (d) Estimate the probability that a person who comes into the store will buy nothing.

In a sales effectiveness seminar, a group of sales representatives tried two approaches to selling a customer a new automobile: the aggressive approach and the passive approach. For 1160 customers, the following record was kept: $$\begin{array}{lllr} \hline & \text { Sale } & \text { No Sale } & \text { Row Total } \\ \hline \text { Aggressive } & 270 & 310 & 580 \\ \text { Passive } & 416 & 164 & 580 \\ \text { Column Total } & 686 & 474 & 1160 \\ \hline \end{array}$$ Suppose a customer is selected at random from the 1160 participating customers. Let us use the following notation for events: \(A=\) aggressive approach, \(P a=\) passive approach, \(S=\) sale, \(N=\) no sale. So, \(P(A)\) is the probability that an aggressive approach was used, and so on. (a) Compute \(P(S), P(S \mid A)\), and \(P(S \mid P a)\). (b) Are the events \(S=\) sale and \(P a=\) passive approach independent? Explain. (c) Compute \(P(A\) and \(S)\) and \(P(P a\) and \(S)\). (d) Compute \(P(N)\) and \(P(N \mid A)\). (e) Are the events \(N=\) no sale and \(A=\) aggressive approach independent? Explain. (f) Compute \(P(A\) or \(S)\).

During the Computer Daze special promotion, a customer purchasing a computer and printer is given a choice of three free software packages. There are 10 different software packages from which to select. How many different groups of software packages can be selected?

Are customers more loyal in the East or in the West? The following table is based on information from Trends in the United States, published by the Food Marketing Institute, Washington, D.C. The columns represent length of customer loyalty (in years) at a primary supermarket. The rows represent regions of the United States. $$\begin{array}{lcrrrrrr} \hline & \begin{array}{c} \text { Less Than } \\ 1 \text { Year } \end{array} & \begin{array}{c} 1-2 \\ \text { Years } \end{array} & \begin{array}{c} 3-4 \\ \text { Years } \end{array} & \begin{array}{c} 5-9 \\ \text { Years } \end{array} & \begin{array}{c} 10-14 \\ \text { Years } \end{array} & \begin{array}{c} 15 \text { or More } \\ \text { Years } \end{array} & \begin{array}{l} \text { Row } \\ \text { Total } \end{array} \\ \hline \text { East } & 32 & 54 & 59 & 112 & 77 & 118 & 452 \\ \text { Midwest } & 31 & 68 & 68 & 120 & 63 & 173 & 523 \\ \text { South } & 53 & 92 & 93 & 158 & 106 & 158 & 660 \\ \text { West } & 41 & 56 & 67 & 78 & 45 & 86 & 373 \\ \text { Column Total } & 157 & 270 & 287 & 468 & 291 & 535 & 2008 \\ \hline \end{array}$$ What is the probability that a customer chosen at random (a) has been loyal 10 to 14 years? (b) has been loyal 10 to 14 years, given that he or she is from the East? (c) has been loyal at least 10 years? (d) has been loyal at least 10 years, given that he or she is from the West? (e) is from the West, given that he or she has been loyal less than 1 year? (f) is from the South, given that he or she has been loyal less than 1 year? (g) has been loyal 1 or more years, given that he or she is from the East? (h) has been loyal 1 or more years, given that he or she is from the West? (i) Are the events from the East and loyal 15 or more years independent? Explain.

One professor grades homework by randomly choosing 5 out of 12 homework problems to grade. (a) How many different groups of 5 problems can be chosen from the 12 problems? (b) Probability extension: Jerry did only 5 problems of one assignment. What is the probability that the problems he did comprised the group that was selected to be graded? (c) Silvia did 7 problems. How many different groups of 5 did she complete? What is the probability that one of the groups of 5 she completed comprised the group selected to be graded?