Question 1

Find the indicated probability.
-An unprepared student makes random guesses for the five true-false questions on a quiz. Find the probability that the student gets at least one correct answer.

Accepted Answer

The probability of guessing at least one correct answer is easier to calculate by finding the probability of the opposite event (getting all answers wrong) and subtracting it from 1. The probability of getting a question wrong is 0.5 (since there are two options, true or false), and for five questions, it's $0.5^5 = 0.03125$. Therefore, the probability of getting at least one correct is $1 - 0.03125 = 0.96875$.

Question 2

The data in the following table show the outcomes of the testing of an allergy drug on 500 people. $$\begin{array} { l | c | c | c | c } 
& \begin{array} { c } 
\text { Allergy } \
\text { drug }
\end{array} & \text { Placebo } & \begin{array} { c } 
\text { Control } \
\text { (no treatment) }
\end{array} & \text { Total } \
\hline \text { Improvement } & 145 & 85 & 41 & 271 \
\text { No improvement } & 55 & 115 & 59 & 229 \
\text { Total } & 200 & 200 & 100 & 500
\end{array}$$
-What is the probability that a randomly selected person was given the allergy drug and did not improve?

Accepted Answer

The probability is calculated by dividing the number of people who were given the allergy drug and did not improve (55) by the total number of people (500), which gives 55/500 = 0.11.

Question 3

Find the indicated probability. Express answers as decimals.
-A sample of 4 different calculators is randomly selected from a group containing 45 that are defective and 30 that have no defects. What is the probability that all four of the calculators selected are defective?

Accepted Answer

The probability of selecting 4 defective calculators out of a total of 75 (45 defective + 30 non-defective) is calculated using combinations. The probability is $\frac{{\binom{45}{4}}}{{\binom{75}{4}}}$, which simplifies to approximately 0.1226.

Question 4

The data in the following table show the outcomes of the testing of an allergy drug on 500 people. $$\begin{array} { l | c | c | c | c } 
& \begin{array} { c } 
\text { Allergy } \
\text { drug }
\end{array} & \text { Placebo } & \begin{array} { c } 
\text { Control } \
\text { (no treatment) }
\end{array} & \text { Total } \
\hline \text { Improvement } & 145 & 85 & 41 & 271 \
\text { No improvement } & 55 & 115 & 59 & 229 \
\text { Total } & 200 & 200 & 100 & 500
\end{array}$$
-What is the probability that a randomly selected person either was given the allergy drug and improved?

Accepted Answer

The answer of The data in the following table show...

Question 5

The data in the following table show the outcomes of the testing of an allergy drug on 500 people. $$\begin{array} { l | c | c | c | c } 
& \begin{array} { c } 
\text { Allergy } \
\text { drug }
\end{array} & \text { Placebo } & \begin{array} { c } 
\text { Control } \
\text { (no treatment) }
\end{array} & \text { Total } \
\hline \text { Improvement } & 145 & 85 & 41 & 271 \
\text { No improvement } & 55 & 115 & 59 & 229 \
\text { Total } & 200 & 200 & 100 & 500
\end{array}$$
-What is the probability that a randomly selected person either was given the allergy drug or did not improve?

Accepted Answer

The answer of The data in the following table show...

Question 6

The data in the following table show the outcomes of the testing of an allergy drug on 500 people. $$\begin{array} { l | c | c | c | c } 
& \begin{array} { c } 
\text { Allergy } \
\text { drug }
\end{array} & \text { Placebo } & \begin{array} { c } 
\text { Control } \
\text { (no treatment) }
\end{array} & \text { Total } \
\hline \text { Improvement } & 145 & 85 & 41 & 271 \
\text { No improvement } & 55 & 115 & 59 & 229 \
\text { Total } & 200 & 200 & 100 & 500
\end{array}$$
-What is the probability that a randomly selected person was not given the placebo or improved?

Accepted Answer

The probability that a person was not given the placebo or improved is calculated by adding the probability of not being given the placebo (300 out of 500) and the probability of improvement (271 out of 500), then subtracting the overlap (people who improved but were given the placebo, 85 out of 500). This gives (300 + 271 - 85) / 500 = 486 / 500 = 0.772, which rounds to 0.77.

Question 7

Find the indicated probability. Express answers as decimals.
-In one town, 68% of adults have health insurance. What is the probability that 5 adults selected at random from the town all have health insurance?

Accepted Answer

The answer of Find the indicated probability. Express answers as...

Question 8

Find the indicated probability. Express answers as decimals.
-A telephone pollster has the names and telephone numbers of 50 voters, 20 of whom are registered Democrats and 30 of whom are registered Republicans. After each call, she removes that person's name from her list. What
Is the probability that the first two calls are to Democrats?

Accepted Answer

The probability of the first call being to a Democrat is 20/50 or 0.4. After removing one Democrat from the list, the probability of the second call being to a Democrat is 19/49. Multiplying these probabilities gives 0.4 * (19/49) = 0.155.

Question 9

Find the indicated probability. Express answers as decimals.
-An IRS auditor randomly selects 3 tax returns from 57 returns of which 15 contain errors. What is the probability that she selects none of those containing errors?

Accepted Answer

The probability of selecting none of the returns containing errors can be calculated using combinations. The total number of ways to select 3 returns out of 57 is 57C3. The number of ways to select 3 returns that do not contain errors (from the 42 that don't contain errors) is 42C3. The probability is therefore (42C3) / (57C3) = 0.3923.

Question 10

Find the indicated probability.
-In a batch of 8000 clock radios 6% are defective. A sample of 12 clock radios is randomly selected without replacement from the 8000 and tested. The entire batch will be rejected if at least one of those tested is defective.
What is the probability that the entire batch will be rejected?

Accepted Answer

To find the probability that the entire batch will be rejected, we first calculate the probability that none of the 12 radios tested is defective, and then subtract that probability from 1. There are 8000 radios in total, with 6% defective, so 480 are defective (6% of 8000) and 7520 are not defective. The probability that the first radio is not defective is 7520/8000, the second is 7519/7999, and so on, down to 7509/7989 for the twelfth radio. The probability that all 12 tested radios are not defective is the product of these probabilities. The probability that at least one is defective (and thus the batch is rejected) is 1 minus this product. Calculating this gives a probability greater than 0.5, making A the correct choice.

Question 11

Find the indicated probability.
-A study conducted at a certain college shows that 57% of the school's graduates find a job in their chosen field within a year after graduation. Find the probability that among 5 randomly selected graduates, at least one
Finds a job in his or her chosen field within a year of graduating.

Accepted Answer

The probability that at least one graduate finds a job is 1 minus the probability that none of the graduates find a job. The probability that a single graduate does not find a job is 0.43 (1 - 0.57). For 5 graduates, this probability is 0.43^5. Thus, the probability that at least one finds a job is 1 - 0.43^5 = 0.985.

Question 12

The data in the following table show the outcomes of the testing of an allergy drug on 500 people. $$\begin{array} { l | c | c | c | c } 
& \begin{array} { c } 
\text { Allergy } \
\text { drug }
\end{array} & \text { Placebo } & \begin{array} { c } 
\text { Control } \
\text { (no treatment) }
\end{array} & \text { Total } \
\hline \text { Improvement } & 145 & 85 & 41 & 271 \
\text { No improvement } & 55 & 115 & 59 & 229 \
\text { Total } & 200 & 200 & 100 & 500
\end{array}$$
-What is the probability that a randomly selected person either was given the allergy drug or improved?

Accepted Answer

The probability of being given the allergy drug is 200/500 = 0.4. The probability of improvement is 271/500 = 0.542. The probability of being given the allergy drug or improving is 0.4 + 0.542 - 0.4 * 0.542 = 0.652.

Question 13

Find the indicated probability. Express answers as decimals.
-You are dealt two cards successively (without replacement)from a shuffled deck of 52 playing cards. Find the probability that both cards are red.

Accepted Answer

There are 26 red cards in a deck of 52 cards. The probability that the first card is red is 26/52. If the first card is red, there are now 25 red cards left out of 51 total cards. The probability that the second card is red is then 25/51. The probability that both cards are red is (26/52) * (25/51) = 0.245.

Question 14

Find the indicated probability. Express answers as decimals.
-You are dealt two cards successively (without replacement)from a shuffled deck of 52 playing cards. Find the probability that the first card is a king and the second card is a queen.

Accepted Answer

The probability of drawing a king first is 4/52 (since there are 4 kings in a deck of 52 cards). After drawing a king, there are 51 cards left, and the probability of then drawing a queen is 4/51 (since there are 4 queens left in the deck). Multiplying these probabilities gives (4/52) * (4/51) = 1/13 * 1/51 = 1/663 ≈ 0.006.

Question 15

The data in the following table show the outcomes of the testing of an allergy drug on 500 people. $$\begin{array} { l | c | c | c | c } 
& \begin{array} { c } 
\text { Allergy } \
\text { drug }
\end{array} & \text { Placebo } & \begin{array} { c } 
\text { Control } \
\text { (no treatment) }
\end{array} & \text { Total } \
\hline \text { Improvement } & 145 & 85 & 41 & 271 \
\text { No improvement } & 55 & 115 & 59 & 229 \
\text { Total } & 200 & 200 & 100 & 500
\end{array}$$
-What is the probability that a randomly selected person either was given the placebo or
Wasn't given the placebo?

Accepted Answer

The probability that a randomly selected person either was given the placebo or wasn't given the placebo includes all participants, which is 100% or a probability of 1.

Question 16

Find the indicated probability. Express answers as decimals.
-You are dealt two cards successively (without replacement)from a shuffled deck of 52 playing cards. Find the probability that both cards are jacks.

Accepted Answer

There are 4 jacks in a deck of 52 cards. The probability of drawing a jack on the first draw is 4/52. After the first draw, there are 3 jacks left in a deck of 51 cards. The probability of drawing a jack on the second draw is 3/51. The probability of drawing two jacks is (4/52) * (3/51) = 0.005.

Question 17

Find the indicated probability.
-In a blood testing procedure, blood samples from 5 people are combined into one mixture. The mixture will only test negative if all the individual samples are negative. If the probability that an individual sample tests positive is 0.08, what is the probability that the mixture will test positive?

Accepted Answer

The probability that an individual sample tests negative is 1 - 0.08 = 0.92. The probability that all 5 samples test negative is 0.92^5. Therefore, the probability that the mixture tests positive is 1 - 0.92^5 = 0.341.

Question 18

Find the indicated probability. Express answers as decimals.
-A study conducted at a certain college shows that 72% of the school's graduates find a job in their chosen field within a year after graduation. Find the probability that 5 randomly selected graduates all find jobs in their chosen field within a year of graduating.

Accepted Answer

The answer of Find the indicated probability. Express answers as...

Question 19

Find the indicated probability. Express answers as decimals.
-Among the contestants in a competition are 35 women and 25 men. If 5 winners are randomly selected, what is the probability that they are all men?

Accepted Answer

The probability of selecting 5 men out of a total of 60 contestants (35 women + 25 men) without replacement is calculated using combinations. The probability is the number of ways to choose 5 men out of 25 divided by the number of ways to choose 5 people out of 60: $P = \frac{{25 \choose 5}}{{60 \choose 5}} = 0.00973$.

Quiz 6: Probability in Statistics

Find the indicated probability. -An unprepared student makes random guesses for the five true-false questions on a quiz. Find the probability that the student gets at least one correct answer.

Find the indicated probability. Express answers as decimals. -A sample of 4 different calculators is randomly selected from a group containing 45 that are defective and 30 that have no defects. What is the probability that all four of the calculators selected are defective?

Find the indicated probability. Express answers as decimals. -In one town, 68% of adults have health insurance. What is the probability that 5 adults selected at random from the town all have health insurance?

Find the indicated probability. Express answers as decimals. -An IRS auditor randomly selects 3 tax returns from 57 returns of which 15 contain errors. What is the probability that she selects none of those containing errors?

Find the indicated probability. Express answers as decimals. -You are dealt two cards successively (without replacement) from a shuffled deck of 52 playing cards. Find the probability that both cards are red.

Find the indicated probability. Express answers as decimals. -You are dealt two cards successively (without replacement) from a shuffled deck of 52 playing cards. Find the probability that the first card is a king and the second card is a queen.

Find the indicated probability. Express answers as decimals. -You are dealt two cards successively (without replacement) from a shuffled deck of 52 playing cards. Find the probability that both cards are jacks.

Find the indicated probability. Express answers as decimals. -Among the contestants in a competition are 35 women and 25 men. If 5 winners are randomly selected, what is the probability that they are all men?