## Try It Now

Work these exercises to see how well you understand this material.

### Exercises

- For two events
*A*and*B*,*P*(*A*) = 0.73,*P*(*B*) = 0.48, and*P*(*A*∩*B*) = 0.29.- Find
*P*(*A*|*B*). - Find
*P*(*B*|*A*). - Determine whether or not A and B are independent.

- Find
- For independent events
*A*and*B*,*P*(*A*) = 0.81 and*P*(*B*) = 0.27.

- Find
*P*(*A*∩*B*). - Find
*P*(*A*|*B*). - Find
*P*(*B*|*A*).

- Find
- For mutually exclusive events
*A*and*B*,*P*(*A*) = 0.17 and*P*(*B*) = 0.32.

- Find
*P*(*A*|*B*). - Find
*P*(*B*|*A*).

- Find
- Compute the following probabilities in connection with the roll of a single fair die.

- The probability that the roll is even.
- The probability that the roll is even, given that it is not a two.
- The probability that the roll is even, given that it is not a one.

- A special deck of 16 cards has 4 that are blue, 4 yellow, 4 green, and 4 red. The four cards of each color are numbered from one to four. A single card is drawn at random. Find the following probabilities.

- The probability that the card drawn is red.
- The probability that the card is red, given that it is not green.
- The probability that the card is red, given that it is neither red nor yellow.
- The probability that the card is red, given that it is not a four.

- A random experiment gave rise to the two-way contingency table shown. Use it to compute the probabilities indicated.

*R**S**A*0.13 0.07 *B*0.61 0.19

*P*(*A*),*P*(*R*),*P*(*A*∩*R*).- Based on the answer to (a), determine whether or not the events
*A*and*R*are independent. - Based on the answer to (b), determine whether or not
*P*(*A*|*R*) can be predicted without any computation. If so, make the prediction. In any case, compute*P*(*A*|*R*) using the Rule for Conditional Probability.

- Suppose for events
*A*and*B*in a random experiment*P*(*A*) = 0.70 and*P*(*B*) = 0.30. Compute the indicated probability, or explain why there is not enough information to do so.

*P*(*A*∩*B*).*P*(*A*∩*B*), with the extra information that*A*and*B*are independent.*P*(*A*∩*B*), with the extra information that*A*and*B*are mutually exclusive.

- Suppose for events
*A, B*, and*C*connected to some random experiment,*A, B*, and*C*are independent and*P*(*A*) = 0.88,*P*(*B*) = 0.65, and*P*(*C*) = 0.44. Compute the indicated probability, or explain why there is not enough information to do so.*P*(*A*∩*B*∩*C*)*P*(*A*∩^{c}*B*∩^{c}*C*)^{c}

- The sample space that describes all three-child families according to the genders of the children with respect to birth order is:
*S*= {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}

In the experiment of selecting a three-child family at random, compute each of the following probabilities, assuming all outcomes are equally likely.- The probability that the family has at least two boys.
- The probability that the family has at least two boys, given that not all of the children are girls.
- The probability that at least one child is a boy.
- The probability that at least one child is a boy, given that the first born is a girl.

- The following two-way contingency table gives the breakdown of the population in a particular locale according to party affiliation (
*A, B, C*, or*None*) and opinion on a bond issue:

Affiliation Opinion Favors Opposes Undecided *A*0.12 0.09 0.07 *B*0.16 0.12 0.14 *C*0.04 0.03 0.06 *None*0.08 0.06 0.03

A person is selected at random. Find each of the following probabilities.- The person is in favor of the bond issue.
- The person is in favor of the bond issue, given that he is affiliated with party
*A*. - The person is in favor of the bond issue, given that he is affiliated with party
*B*.

- The following two-way contingency table gives the breakdown of the population of adults in a particular locale according to employment type and level of life insurance:

Employment Type Level of Insurance Low Medium High Unskilled 0.07 0.19 0.00 Semi-skilled 0.04 0.28 0.08 Skilled 0.03 0.18 0.05 Professional 0.01 0.05 0.02

An adult is selected at random. Find each of the following probabilities.- The person has a high level of life insurance.
- The person has a high level of life insurance, given that he does not have a professional position.
- The person has a high level of life insurance, given that he has a professional position.
- Determine whether or not the events "has a high level of life insurance" and "has a professional position" are independent.

- The
*sensitivity*of a drug test is the probability that the test will be positive when administered to a person who has actually taken the drug. Suppose that there are two independent tests to detect the presence of a certain type of banned drugs in athletes. One has sensitivity 0.75; the other has sensitivity 0.85. If both are applied to an athlete who has taken this type of drug, what is the chance that his usage will go undetected? - An accountant has observed that 5% of all copies of a particular two-part form have an error in Part I, and 2% have an error in Part II. If the errors occur independently, find the probability that a randomly selected form will be error-free.
- Events
*A*and*B*are mutually exclusive. Find*P*(*A*|*B*). - A basketball player makes 60% of the free throws that he attempts, except that if he has just tried and missed a free throw then his chances of making a second one go down to only 30%. Suppose he has just been awarded two free throws.
- Find the probability that he makes both.
- Find the probability that he makes at least one. (A tree diagram could help.)

Source: https://saylordotorg.github.io/text_introductory-statistics/index.html

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