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INTRODUCTION TO PROBABILITY Answer the following probability problems showing your work and explaining (or analyzing) your results.

1. In a poll, respondents were asked if they have traveled to Europe. 68 respondents indicated that they have traveled to Europe and 124 respondents said that they have not traveled to Europe. If one of these respondents is randomly selected, what is the probability of getting someone who has traveled to Europe?

Solution

To get that probability we take the number of people who have visited Europe divided by the total number of respondents.

p: 68/(68+124) = 68/192 = 0.3542

2. The data set represents the income levels of the members of a golf club. Find the probability that a randomly selected member earns at least $100,000.

INCOME (in thousands of dollars)

98 102 83 140 201 96 74 109 163 210

81 104 134 158 128 107 87 79 91 121

Solution

There are 20 members in total and 12 of them earn at least $100,000. So the probability that a randomly selected member earns at least $100,000 is:

P= 12/20= 3/5 or 0.6

3. A poll was taken to determine the birthplace of a class of college students. Below is a chart of the results.

Solutions

a. What is the probability that a female student was born in Orlando?

P= Number of favorable events ÷total number of events

Total number of students= 10+16+5+12+7+9= 59

P (female born in Orlando) = 12/59 or 0.2034

b. What is the probability that a male student was born in Miami?

P (male born in Miami) = 7/59 or 0.1186

c. What is the probability that a student was born in Jacksonville?

P (student born in Jacksonville)= (10+16)/29 = 26/59 or 0.4407

Gender

Number of students

Location of birth

Male

10

Jacksonville

Female

16

Jacksonville

Male

5

Orlando

Female

12

Orlando

Male

7

Miami

Female

9

Miami

4. Of the 538 people who had an annual check-up at a doctor’s office, 215 had high blood pressure. Estimate the probability that the next person who has a check-up will have high blood pressure.

Solution

In case the first 538 people represents all the people that visited the Doctor’s office for annual check-ups, then the probability of people that come for annual checkups and have high blood pressure is

P= 215/538.

It is approximately 0.3996 or 39.96%

This could be rounded off to 0.4 or 40%.

Because 0.4 = 2/5, it can be said that the next individual has a 2 in 5 chance/likelihood of having high blood pressure.

5. Find the probability of correctly answering the first 4 questions on a multiple choice test using random guessing. Each question has 3 possible answers.

Solution

The probability of answering a question right is 1/3 since there are three possible answers. Since there are four questions, we need to multiply 1/3 by 4 to get the probability of answering the first four questions

P= 1/3*1/3*1/3*1/3 or 1/3*^4= 1/81

6. Explain the difference between an independent and a dependent variable.

In independent variable is the variable that is often controlled or change in a scientific experiment so as to test the effects or impacts on the dependent variable while a dependent variable is the variable that is being measured or tested in a scientific experiment.

7. Provide an example of experimental probability and explain why it is considered experimental.

An example of an experimental probability is rolling dice 10 times. Suppose number ‘5’ occurs 4 times, the experimental probability is 4/10= 2/5 or 0.4. This is because:

Experimental probability = Number of event occurrence ÷ total number of trials.

Rolling dice 10 times is considered a theoretical probability because it is the probability of an event (number ‘5’) occurring when an experiment is conducted (rolling the dice).

8. The measure of how likely an event will occur is probability. Match the following probability with one of the statements. There is only one answer per statement.

0 0.25 0.60 1

Solutions

a. This event is certain and will happen every time. P=1

Probability is a measure of how likely an event will occur

A certain event has a probability of 1

b. This event will happen more often than not. P= 0.25

An event that is not likely has a probability of nearly 0.25.

c. This event will never happen. P= 0

An impossible event has zero probability of occurring.

d. This event is likely and will occur occasionally. P= 0.60.

A likely event has a probability of nearly 0.75

9. Flip a coin 25 times and keep track of the results. What is the experimental probability of landing on tails? What is the theoretical probability of landing on heads or tails?

Solutions

Experimental probability

After flipping the coin 25 times, I had 15 heads and 10 tails.

The experimental probability of the coin landing on tails is therefore,

P= Number of favorable events ÷total number of events

P= 10/25= 2/5 or 0.4

Theoretical probability

Since the coin has a head and a tail, the theoretical probability of the coin landing on tails (1/2 or ½) and landing on heads (1/2 or 0.5)

But the theoretical probability of landing on heads or tails = 25/25= 1

10. A color candy was chosen randomly out of a bag. Below are the results:

Color

Probability

Blue

0.30

Red

0.10

Green

0.15

Yellow

0.20

Orange

???

Solutions

a. What is the probability of choosing a yellow candy?

P (yellow candy) = 0.20

b. What is the probability that the candy is blue, red, or green?

The probability that the candy is blue, red, or green is the sum of their individual probabilities.

P (blue, red, or green) = 0.3+0.1+0.15) = 0.55

c. What is the probability of choosing an orange candy?

The total probability is always one (1), so the probability of choosing orange is 1- the total probability of the other colors.

P (orange) = 1-(0.30+0.10+0.20+ 0.15)

P (orange) = 1-0.75= 0.25

1. In a poll, respondents were asked if they have traveled to Europe. 68 respondents indicated that they have traveled to Europe and 124 respondents said that they have not traveled to Europe. If one of these respondents is randomly selected, what is the probability of getting someone who has traveled to Europe?

Solution

To get that probability we take the number of people who have visited Europe divided by the total number of respondents.

p: 68/(68+124) = 68/192 = 0.3542

2. The data set represents the income levels of the members of a golf club. Find the probability that a randomly selected member earns at least $100,000.

INCOME (in thousands of dollars)

98 102 83 140 201 96 74 109 163 210

81 104 134 158 128 107 87 79 91 121

Solution

There are 20 members in total and 12 of them earn at least $100,000. So the probability that a randomly selected member earns at least $100,000 is:

P= 12/20= 3/5 or 0.6

3. A poll was taken to determine the birthplace of a class of college students. Below is a chart of the results.

Solutions

a. What is the probability that a female student was born in Orlando?

P= Number of favorable events ÷total number of events

Total number of students= 10+16+5+12+7+9= 59

P (female born in Orlando) = 12/59 or 0.2034

b. What is the probability that a male student was born in Miami?

P (male born in Miami) = 7/59 or 0.1186

c. What is the probability that a student was born in Jacksonville?

P (student born in Jacksonville)= (10+16)/29 = 26/59 or 0.4407

Gender

Number of students

Location of birth

Male

10

Jacksonville

Female

16

Jacksonville

Male

5

Orlando

Female

12

Orlando

Male

7

Miami

Female

9

Miami

4. Of the 538 people who had an annual check-up at a doctor’s office, 215 had high blood pressure. Estimate the probability that the next person who has a check-up will have high blood pressure.

Solution

In case the first 538 people represents all the people that visited the Doctor’s office for annual check-ups, then the probability of people that come for annual checkups and have high blood pressure is

P= 215/538.

It is approximately 0.3996 or 39.96%

This could be rounded off to 0.4 or 40%.

Because 0.4 = 2/5, it can be said that the next individual has a 2 in 5 chance/likelihood of having high blood pressure.

5. Find the probability of correctly answering the first 4 questions on a multiple choice test using random guessing. Each question has 3 possible answers.

Solution

The probability of answering a question right is 1/3 since there are three possible answers. Since there are four questions, we need to multiply 1/3 by 4 to get the probability of answering the first four questions

P= 1/3*1/3*1/3*1/3 or 1/3*^4= 1/81

6. Explain the difference between an independent and a dependent variable.

In independent variable is the variable that is often controlled or change in a scientific experiment so as to test the effects or impacts on the dependent variable while a dependent variable is the variable that is being measured or tested in a scientific experiment.

7. Provide an example of experimental probability and explain why it is considered experimental.

An example of an experimental probability is rolling dice 10 times. Suppose number ‘5’ occurs 4 times, the experimental probability is 4/10= 2/5 or 0.4. This is because:

Experimental probability = Number of event occurrence ÷ total number of trials.

Rolling dice 10 times is considered a theoretical probability because it is the probability of an event (number ‘5’) occurring when an experiment is conducted (rolling the dice).

8. The measure of how likely an event will occur is probability. Match the following probability with one of the statements. There is only one answer per statement.

0 0.25 0.60 1

Solutions

a. This event is certain and will happen every time. P=1

Probability is a measure of how likely an event will occur

A certain event has a probability of 1

b. This event will happen more often than not. P= 0.25

An event that is not likely has a probability of nearly 0.25.

c. This event will never happen. P= 0

An impossible event has zero probability of occurring.

d. This event is likely and will occur occasionally. P= 0.60.

A likely event has a probability of nearly 0.75

9. Flip a coin 25 times and keep track of the results. What is the experimental probability of landing on tails? What is the theoretical probability of landing on heads or tails?

Solutions

Experimental probability

After flipping the coin 25 times, I had 15 heads and 10 tails.

The experimental probability of the coin landing on tails is therefore,

P= Number of favorable events ÷total number of events

P= 10/25= 2/5 or 0.4

Theoretical probability

Since the coin has a head and a tail, the theoretical probability of the coin landing on tails (1/2 or ½) and landing on heads (1/2 or 0.5)

But the theoretical probability of landing on heads or tails = 25/25= 1

10. A color candy was chosen randomly out of a bag. Below are the results:

Color

Probability

Blue

0.30

Red

0.10

Green

0.15

Yellow

0.20

Orange

???

Solutions

a. What is the probability of choosing a yellow candy?

P (yellow candy) = 0.20

b. What is the probability that the candy is blue, red, or green?

The probability that the candy is blue, red, or green is the sum of their individual probabilities.

P (blue, red, or green) = 0.3+0.1+0.15) = 0.55

c. What is the probability of choosing an orange candy?

The total probability is always one (1), so the probability of choosing orange is 1- the total probability of the other colors.

P (orange) = 1-(0.30+0.10+0.20+ 0.15)

P (orange) = 1-0.75= 0.25