Exam 6: Some Rules of Probability

In an experiment, persons are asked to pick a number from 10 to 18 , so that for each person the sample space is the set $S=\{10,11,12,13,14,15,16,17,18\}$ . If $A=\{10,11,15,16,17\}, B=\{10,12,14\}$ , and $C=\{14,16,18\}$ , list the elements of the sample space comprising each of the following events. - $B \cup C$

A company has discovered a way of evaluating the success of both their radio and television advertising. If $R$ and $T$ are, respectively, the events that the radio advertising and television advertising are successful, $P(R)=0.62, P(T)=0.75$ , and $P(R \cap T)=\mathbf{0 . 4 3}$ . -Using the situation above, state in words what probability is expressed by the following: $P\left(R \cap T^{\prime}\right)$ .

The expressions $A^{\prime} \cap B^{\prime}$ and $(A \cap B)^{\prime}$ are equal.

A school has tabulated the favorite snacks of 1000 of its students in the two categories, males and females. Here are the results: If one of the terms in the table above is selected at random, find each of the following probabilities. - $P\left(I^{\prime}\right)=$ _______

Two probabilities may be added when we are asked

A company has discovered a way of evaluating the success of both their radio and television advertising. If $R$ and $T$ are, respectively, the events that the radio advertising and television advertising are successful, $P(R)=0.62, P(T)=0.75$ , and $P(R \cap T)=\mathbf{0 . 4 3}$ . -Using the situation above, state in words what probability is expressed by the following: $P\left(R^{\prime}\right)$

Thirty percent of students attending a certain student mixer meet someone new to date. Forty percent of students attending the mixer dance at sometime during the mixer. Of those who dance, $60 \%$ meet someone new to date. A student who attends the mixer is randomly selected. -In the situation above, find the probability that he/she either dances or meets someone new to date.

A basketball coach plans to add two players from among five juniors and eight seniors. What is the probability that -the first will be a junior and the second will be a senior?

Given mutually exclusive events $C$ and $D$ for which $P(C)=0.61$ and $P(D)=0.34$ , find - $P\left(C^{\prime} \cup D^{\prime}\right)$ .

A school has tabulated the favorite snacks of 1000 of its students in the two categories, males and females. Here are the results: If one of the terms in the table above is selected at random, find each of the following probabilities. - $P(F \mid I)=$ _______

A company estimates that the probability of a recession occurring in the next year is 0.4 . The company also estimates the probability that another company distributes a competing product in the next year is 0.5 . Finally, the company feels that the probability of both a recession occurring and a competing product being produced in the next year is $\mathbf{0 . 2 5}$ . -In the situation above, if there is a recession, what is the probability that a company will distribute a competing product in the next year?

Two options an automobile buyer may purchase are air-conditioning $(C)$ and an automatic transmission $(T)$ . A dealer notes from his sales records that the probability of a buyer purchasing an automatic transmission is 0.60 and the probability that he purchased air-conditioning is 0.50 . The probability that the buyer bought air-conditioning if he bought an automatic transmission is 0.70 . -In the situation above, find the probability that a buyer did not purchase either air-conditioning or an automatic transmission.

If $A$ and $B$ are mutually exclusive events with $P(A)=0.15, P(B)=0.45$ , then $P\left\{(A \cup B)^{\prime}\right\}$ equals _______.

A consumer has placed two orders for a new product from two different suppliers $X$ and $Y$ . The probabilities that the suppliers deliver the product on time are 0.40 for $X$ and 0.60 for $Y$ .45) If the probability of one supplier delivering the product on time has no effect on whether or not the other one does, find the probability that one or both of the suppliers will deliver the product in the required time. -If the probability of one supplier delivering the product on time has no effect on whether or not the other one does, find the probability that one or both of the suppliers will deliver the product in the required time.

If $C$ is the event that a student buys a stereo and $D$ is the event that the student buys a personal computer, then $P\left(C \cup D^{\prime}\right)$ can be described in words as the probability that _______.

-Use the data in Table 3 to solve the following: The probability expressed by $P(S \cup W)$ equals

The expressions $A \cup B$ and $(A \cap B)^{\prime}$ are equal.

Three families A, B, and C, are bidding on the same one -family house. The probabilities are $0.20,0.25$ , and 0.28 , respectively, that a given family eventually moves into the house. -In the situation above, if $A$ is the event that family $A$ moves into the house, and $B$ is the event that family $B$ moves into the house, the events $\mathrm{A}$ and $\mathrm{B}$ are

The probability of $A$ given $B$ is expressed as $P(A \mid B)$ .

Thirty percent of students attending a certain student mixer meet someone new to date. Forty percent of students attending the mixer dance at sometime during the mixer. Of those who dance, $60 \%$ meet someone new to date. A student who attends the mixer is randomly selected. -In the situation above, find the probability that he/she has danced if you know that he/she met someone new to date.