Question 1

Solve the problem.
-A traffic light at an intersection has a 120-sec cycle. The light is green for 80 sec, yellow for 5 sec, and red for 35 sec.
a. When a motorist approaches the intersection, find the probability that the light will be red.
(Assume that the color of the light is defined as the color when the car is 100 ft from the Intersection. This is the approximate distance at which the driver makes a decision to stop or go.)
b. If a motorist approaches the intersection twice during the day, find the probability that the light
Will be red both times.

Accepted Answer

a. The probability that the light will be red is the duration of the red light divided by the total cycle time, which is $ \frac{35}{120} = \frac{7}{24} $. b. The probability that the light will be red both times a motorist approaches the intersection is the product of the probabilities for each approach, which is $ \left(\frac{7}{24}\right)^2 = \frac{49}{576} $.

Question 2

Solve the problem.
-A basketball player makes approximately 69% of free throws. If she plays in a game in which she shoots 7 free throws, what is the probability the she will make all 7?

Accepted Answer

The answer of Solve the problem.
-A basketball player makes approximately...

Question 3

Solve the problem.
-A test has 11 questions. Seven questions are true/false and four are multiple-choice. Each multiple-choice question has 3 possible responses of which exactly one is correct. Find the
Probability that a student guesses on each question and gets a perfect score.

Accepted Answer

The probability of guessing all 7 true/false questions correctly is $ \left(\frac{1}{2}\right)^7 $. The probability of guessing all 4 multiple-choice questions correctly is $ \left(\frac{1}{3}\right)^4 $. The combined probability is $ \left(\frac{1}{2}\right)^7 \times \left(\frac{1}{3}\right)^4 = \frac{1}{128} \times \frac{1}{81} = \frac{1}{10,368} $.

Question 4

Use the data in the table categorizing cholesterol levels by the ages of the individuals in a study. If one person from the study is chosen at random, find the probability of the given event.
-The person is between 31 and 60, inclusive, or elevated cholesterol. $$\begin{array} { | l | c | c | c | } 
\hline & \begin{array} { c } 
\text { Normal } \
\text { Cholesterol }
\end{array} & \begin{array} { c } 
\text { Elevated } \
\text { Cholesterol }
\end{array} & \text { Total } \
\hline \begin{array} { l } 
\text { 30 and } \
\text { under }
\end{array} & 15 & 3 & 18 \
\hline \mathbf { 3 1 - 6 0 } & 51 & 30 & 81 \
\hline \mathbf { 6 1 } \text { or older } & 19 & 82 & 101 \
\hline \text { Total } & 85 & 115 & 200 \
\hline
\end{array}$$

Accepted Answer

The probability of the person being between 31 and 60, inclusive, is 81/200. The probability of the person having elevated cholesterol is 115/200. The probability of the person being between 31 and 60, inclusive, or having elevated cholesterol is (81/200) + (115/200) - (30/200) = 83/100.

Question 5

Solve the problem.
-A slot machine in a casino has four wheels that all spin independently. Each wheel has 11 stops, denoted by 0 through 9, and bar. What is the probability that a given outcome is bar-bar-bar?

Accepted Answer

Since each wheel has 11 stops and they spin independently, the total number of possible outcomes is $11^3 = 1,331$. The desired outcome, bar-bar-bar, represents only one of these possible outcomes. Therefore, the probability is $\frac{1}{1,331}$, which matches choice A.

Question 6

Use the data in the table categorizing cholesterol levels by the ages of the individuals in a study. If one person from the study is chosen at random, find the probability of the given event.
-The person is 60 or under. $\begin{array}{|l|c|c|c|}
\hline & \begin{array}{c}
\text { Normal } \
\text { Cholesterol }
\end{array} & \begin{array}{c}
\text { Elevated } \
\text { Cholesterol }
\end{array} & \text { Total } \
\hline \begin{array}{l}
\text { 30 and } \
\text { under }
\end{array} & 17 & 4 & 21 \
\hline \text { 31-60 } & 49 & 32 & 81 \
\hline \text { 61 or older } & 22 & 76 & 98 \
\hline \text { Total } & 88 & 112 & 200 \
\hline
\end{array}$

Accepted Answer

The total number of people who are 60 or under is 21 (30 and under) + 81 (31-60) = 102. The probability is 102/200 = 51/100.

Question 7

Solve the problem.
-Airlines often overbook flights because a small percentage of passengers do not show up (perhaps due to missed connections). Past history indicates that for a certain route, the probability that an
Individual passenger will not show up is 0.03. Suppose that 66 people bought tickets for a flight that
Has 65 seats. Determine the probability that there will not be enough seats. Round to 4 decimal
Places.

Accepted Answer

The probability that a passenger will show up is 1 - 0.03 = 0.97. The probability that all 66 passengers will show up is 0.97^66 = 0.1741. The probability that there will not be enough seats is 1 - 0.1741 = 0.1339.

Question 8

Solve the problem.
-Suppose a die is rolled followed by the flip of a coin. Find the probability that the outcome is a 3 on the die followed by the coin turning up heads.

Accepted Answer

The probability of rolling a 3 on a die is $\frac{1}{6}$ (since there are 6 sides on a die), and the probability of flipping heads on a coin is $\frac{1}{2}$. Since these are independent events, you multiply the probabilities: $\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$.

Question 9

Use the data in the table categorizing cholesterol levels by the ages of the individuals in a study. If one person from the study is chosen at random, find the probability of the given event.
-The person has normal cholesterol or is 61 or older. $\begin{array}{|l|c|c|c|}
\hline & \begin{array}{c}
\text { Normal } \
\text { Cholesterol }
\end{array} & \begin{array}{c}
\text { Elevated } \
\text { Cholesterol }
\end{array} & \text { Total } \
\hline \begin{array}{l}
\mathbf{3 0} \text { and } \
\text { under }
\end{array} & 11 & 5 & 16 \
\hline \text { 31-60 } & 45 & 31 & 76 \
\hline \text { 61 or older } & 18 & 90 & 108 \
\hline \text { Total } & 74 & 126 & 200 \
\hline
\end{array}$

Accepted Answer

The probability of selecting someone with normal cholesterol or 61 or older is the sum of the probabilities of each individual event, minus the overlap (since people 61 or older with normal cholesterol are counted in both categories). There are 74 people with normal cholesterol and 108 people who are 61 or older. However, 18 of those 61 or older have normal cholesterol, which we've already counted in the 74. So, we don't subtract anything because we're looking for the union of both events, not the intersection. Therefore, the total number of people who have normal cholesterol or are 61 or older is 74 (normal cholesterol) + 90 (61 or older with elevated cholesterol) = 164. The total number of people in the study is 200. Thus, the probability is $ \frac{164}{200} = \frac{41}{50} $.

Question 10

Solve the problem.
-The 5-yr survival rate for a type of cancer is 84%. If two people with this type of cancer are selected at random, what is the probability that they both survive 5 yr?

Accepted Answer

The probability that both survive is found by multiplying the individual probabilities together: 0.84 * 0.84 = 0.7056.

Quiz 8: Sequences, Series, Induction, and Probability

Solve the problem. -A basketball player makes approximately 69% of free throws. If she plays in a game in which she shoots 7 free throws, what is the probability the she will make all 7?

Solve the problem. -A test has 11 questions. Seven questions are true/false and four are multiple-choice. Each multiple-choice question has 3 possible responses of which exactly one is correct. Find the Probability that a student guesses on each question and gets a perfect score.

Solve the problem. -A slot machine in a casino has four wheels that all spin independently. Each wheel has 11 stops, denoted by 0 through 9, and bar. What is the probability that a given outcome is bar-bar-bar?

Solve the problem. -Suppose a die is rolled followed by the flip of a coin. Find the probability that the outcome is a 3 on the die followed by the coin turning up heads.

Solve the problem. -The 5-yr survival rate for a type of cancer is 84%. If two people with this type of cancer are selected at random, what is the probability that they both survive 5 yr?