# 3-3. Exercises

## 1. **BASIC**

**Ex 1.** For two events $$A$$ and $$B$$ , $$P(A)=0.73, P(B)=0.48$$ , and $$P(A∩B)=0.29$$ .

1. Find $$P(A|B).$$&#x20;
2. Find $$P(B|A).$$&#x20;
3. Determine whether or not $$A$$ and $$B$$ are independent.

**Ex 2.** For two events $$A$$ and $$B$$ , $$P(A)=0.26, P(B)=0.37$$ , and $$P(A∩B)=0.11$$ .

1. Find $$P(A|B).$$&#x20;
2. Find $$P(B|A).$$&#x20;
3. Determine whether or not $$A$$ and $$B$$ are independent.

**Ex 3.** For two events $$A$$ and $$B$$ , $$P(A)=0.81,$$ and $$P(B)=0.27$$ .

1. Find $$P(A∩B).$$&#x20;
2. Find $$P(A|B).$$&#x20;
3. Find $$P(B|A).$$&#x20;

**Ex 4.** For two events $$A$$ and $$B$$ , $$P(A)=0.68,$$ and $$P(B)=0.37$$ .

1. Find $$P(A∩B).$$
2. Find $$P(A|B).$$&#x20;
3. Find $$P(B|A).$$&#x20;

**Ex 5.** For mutually exclusive events $$A$$ and $$B$$ , $$P(A)=0.17,$$ and $$P(B)=0.32$$ .

1. Find $$P(A|B).$$&#x20;
2. Find $$P(B|A).$$&#x20;

**Ex 6.** For mutually exclusive events $$A$$ and $$B$$ , $$P(A)=0.45,$$ and $$P(B)=0.09$$ .

1. Find $$P(A|B).$$&#x20;
2. Find $$P(B|A).$$&#x20;

**Ex 7.** Compute the following probabilities in connection with the roll of a single fair die.

1. The probability that the roll is even.
2. The probability that the roll is even, given that it is not a two.
3. The probability that the roll is even, given that it is not a one.

**Ex 8.** Compute the following probabilities in connection with two tosses of a fair coin.

1. The probability that the second toss is heads.
2. The probability that the second toss is heads, given that the first toss is heads.
3. The probability that the second toss is heads, given that at least one of the two tosses is heads.

**Ex 9.** 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.

1. The probability that the card drawn is red.
2. The probability that the card is red, given that it is not green.
3. The probability that the card is red, given that it is neither red nor yellow.
4. The probability that the card is red, given that it is not a four.

**Ex 10.** 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.

1. The probability that the card drawn is a two or a four.
2. The probability that the card is a two or a four, given that it is not a one.
3. The probability that the card is a two or a four, given that it is either a two or a three.
4. The probability that the card is a two or a four, given that it is red or green.

**Ex 11.** A random experiment gave rise to the two-way contingency table shown. Use it to compute the probabilities indicated.

|        | $$R$$ | $$S$$  |
| ------ | ----- | ------ |
| $$A$$  | 0.12  | 0.18   |
| $$B$$  | 0.28  | 0.42   |

1. $$P(A), P(R), P(A∩R).$$&#x20;
2. Based on the answer to (a), determine whether or not the events $$A$$ and $$R$$ are independent.
3. 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.

**Ex 12.** 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   |

1. $$P(A), P(R), P(A∩R).$$&#x20;
2. Based on the answer to (a), determine whether or not the events $$A$$ and $$R$$ are independent.
3. 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.

**Ex 13.** 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.

1. $$P(A∩B).$$&#x20;
2. $$P(A∩B)$$ , with the extra information that $$A$$ and $$B$$ are independent.
3. $$P(A∩B)$$ , with the extra information that $$A$$  and $$B$$ are mutually exclusive.

**Ex 14.** Suppose for events _A_ and _B_ connected to some random experiment, $$P(A)=0.50$$ and $$P(B)=0.50$$ . Compute the indicated probability, or explain why there is not enough information to do so.

1. $$P(A∩B).$$&#x20;
2. $$P(A∩B)$$ , with the extra information that $$A$$ and $$B$$ are independent.
3. $$P(A∩B)$$ , with the extra information that $$A$$  and $$B$$ are mutually exclusive.

**Ex 15.** 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.

1. $$P(A∩B∩C)$$&#x20;
2. $$P(A^c∩B^c∩C^c)$$&#x20;

**Ex 16.** Suppose for events $$A$$, $$B$$, and $$C$$ connected to some random experiment, $$A$$, $$B$$, and $$C$$ are independent and $$P(A)=0.95, P(B)=0.73$$ , and $$P(C)=0.62$$ . Compute the indicated probability, or explain why there is not enough information to do so.

1. $$P(A∩B∩C)$$&#x20;
2. $$P(A^c∩B^c∩C^c)$$&#x20;

## **2. APPLICATIONS**

**Ex 17.** The sample space that describes all three-child families according to the genders of the children with respect to birth order is\
\
&#x20;$$S=\{bbb,bbg,bgb,bgg,gbb,gbg,ggb,ggg\}$$ \


In the experiment of selecting a three-child family at random, compute each of the following probabilities, assuming all outcomes are equally likely.

1. The probability that the family has at least two boys.
2. The probability that the family has at least two boys, given that not all of the children are girls.
3. The probability that at least one child is a boy.
4. The probability that at least one child is a boy, given that the first born is a girl.

**Ex 18.** The following two-way contingency table gives the breakdown of the population in a particular locale according to age and number of vehicular moving violations in the past three years:

|          | Violations |      |      |
| -------- | ---------- | ---- | ---- |
| Age      | 0          | 1    | 2+   |
| Under 21 | 0.04       | 0.06 | 0.02 |
| 21–40    | 0.25       | 0.16 | 0.01 |
| 41–60    | 0.23       | 0.10 | 0.02 |
| 60+      | 0.08       | 0.03 | 0.00 |

A person is selected at random. Find the following probabilities.

1. The person is under 21.
2. The person has had at least two violations in the past three years.
3. The person has had at least two violations in the past three years, given that he is under 21.
4. The person is under 21, given that he has had at least two violations in the past three years.
5. Determine whether the events “the person is under 21” and “the person has had at least two violations in the past three years” are independent or not.

**Ex 19.** 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:

|             | Opinion |         |           |
| ----------- | ------- | ------- | --------- |
| Affiliation | 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.

1. The person is in favor of the bond issue.
2. The person is in favor of the bond issue, given that he is affiliated with party $$A$$ .
3. The person is in favor of the bond issue, given that he is affiliated with party $$B$$ .

**Ex 20.** The following two-way contingency table gives the breakdown of the population of patrons at a grocery store according to the number of items purchased and whether or not the patron made an impulse purchase at the checkout counter:

|                 | Impulse Purchase |          |
| --------------- | ---------------- | -------- |
| Number of Items | Made             | Not Made |
| Few             | 0.01             | 0.19     |
| Many            | 0.04             | 0.76     |

A patron is selected at random. Find each of the following probabilities.

1. The patron made an impulse purchase.
2. The patron made an impulse purchase, given that the total number of items purchased was many.
3. Determine whether or not the events “few purchases” and “made an impulse purchase at the checkout counter” are independent.

**Ex 21.** 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:

|                 | Level of Insurance |        |      |
| --------------- | ------------------ | ------ | ---- |
| Employment Type | 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.

1. The person has a high level of life insurance.
2. The person has a high level of life insurance, given that he does not have a professional position.
3. The person has a high level of life insurance, given that he has a professional position.
4. Determine whether or not the events “has a high level of life insurance” and “has a professional position” are independent.

**Ex 22.** The sample space of equally likely outcomes for the experiment of rolling two fair dice is\
\
&#x20;             11  12  13  14  15  16\
&#x20;             21  22  23  24  25  26\
&#x20;             31  32  33  34  35  36\
&#x20;             41  42  43  44  45  46\
&#x20;             51  52  53  54  55  56\
&#x20;             61  62  63  64  65  66\


Identify the events $$N$$ : the sum is at least nine, $$T$$ : at least one of the dice is a two, and $$F$$ : at least one of the dice is a five.

1. Find $$P(N)$$ .
2. Find $$P(N|F).$$&#x20;
3. Find $$P(N|T).$$&#x20;
4. Determine from the previous answers whether or not the events $$N$$ and $$F$$ are independent; whether or not $$N$$ and $$T$$ are.

**Ex 23.** 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?

**Ex 24.** A man has two lights in his well house to keep the pipes from freezing in winter. He checks the lights daily. Each light has probability 0.002 of burning out before it is checked the next day (independently of the other light).

1. If the lights are wired in parallel one will continue to shine even if the other burns out. In this situation, compute the probability that at least one light will continue to shine for the full 24 hours. Note the greatly increased reliability of the system of two bulbs over that of a single bulb.
2. If the lights are wired in series neither one will continue to shine even if only one of them burns out. In this situation, compute the probability that at least one light will continue to shine for the full 24 hours. Note the slightly decreased reliability of the system of two bulbs over that of a single bulb.

**Ex 25.** 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.

**Ex 26.** A box contains 20 screws which are identical in size, but 12 of which are zinc coated and 8 of which are not. Two screws are selected at random, without replacement.

1. Find the probability that both are zinc coated.
2. Find the probability that at least one is zinc coated.

## **3. ADDITIONAL EXERCISES**

**Ex 27.** Events $$A$$ and $$B$$ are mutually exclusive. Find $$P(A|B)$$ .

**Ex 28 .** The city council of a particular city is composed of five members of party $$A$$, four members of party $$B$$, and three independents. Two council members are randomly selected to form an investigative committee.

1. Find the probability that both are from party $$A$$.
2. Find the probability that at least one is an independent.
3. Find the probability that the two have different party affiliations (that is, not both $$A$$, not both $$B$$, and not both independent).

**Ex 29.** 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.

1. Find the probability that he makes both.
2. Find the probability that he makes at least one. (A tree diagram could help.)

**Ex 30.** An economist wishes to ascertain the proportion _p_ of the population of individual taxpayers who have purposely submitted fraudulent information on an income tax return. To truly guarantee anonymity of the taxpayers in a random survey, taxpayers questioned are given the following instructions.

1. Flip a coin.
2. If the coin lands heads, answer “Yes” to the question “Have you ever submitted fraudulent information on a tax return?” even if you have not.
3. If the coin lands tails, give a truthful “Yes” or “No” answer to the question “Have you ever submitted fraudulent information on a tax return?”

The questioner is not told how the coin landed, so he does not know if a “Yes” answer is the truth or is given only because of the coin toss.

1.  Using the Probability Rule for Complements and the independence of the coin toss and the taxpayers’ status fill in the empty cells in the two-way contingency table shown. Assume that the coin is fair. Each cell except the two in the bottom row will contain the unknown proportion (or probability) _p_.

    |             | Coin |   |             |
    | ----------- | ---- | - | ----------- |
    | Status      | H    | T | Probability |
    | Fraud       |      |   | _p_         |
    | No fraud    |      |   |             |
    | Probability |      |   | 1           |
2.  The only information that the economist sees are the entries in the following table:\
    \
    Response          $$“Yes”$$    $$“No”$$ \
    Proportion               $$r$$              $$s$$ \


    Equate the entry in the one cell in the table in (a) that corresponds to the answer “No” to the number $$s$$ to obtain the formula $$p=1−2s$$ that expresses the unknown number $$p$$ in terms of the known number $$s$$ .
3. Equate the sum of the entries in the three cells in the table in (a) that together correspond to the answer “Yes” to the number _r_ to obtain the formula $$p=2r−1$$ that expresses the unknown number _p_ in terms of the known number $$r$$ .
4. Use the fact that $$r+s=1$$ (since they are the probabilities of complementary events) to verify that the formulas in (b) and (c) give the same value for $$p$$ . (For example, insert $$s=1−r$$ into the formula in (b) to obtain the formula in (c).)
5. Suppose a survey of 1,200 taxpayers is conducted and 690 respond “Yes” (truthfully or not) to the question “Have you ever submitted fraudulent information on a tax return?” Use the answer to either (b) or (c) to estimate the true proportion _p_ of all individual taxpayers who have purposely submitted fraudulent information on an income tax return.
