Venn Diagrams and Non-Mutually Exclusive events



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Date: ___________________

Chapter 3 Remediation Packet

Based on your chapter 3 indicator results, you need to practice the following skills before your summative assessment.

|Topic |To be completed |

|Determining Outcomes/Basic Probability | |

|Geometric Probability | |

|Arrangements | |

|Tree Diagrams/Compound Probability | |

|Venn/Diagrams/Mutually Exclusive | |

You must complete at least 3 of the sheets to be graded, with work shown (original, nonsimplified fraction for probability, factorials or numbers that are being multiplied together for arrangements). Answers may be left in fraction in simplest form, decimal to 3 decimal places, or nearest percent, unless otherwise indicated.

Venn Diagrams and Mutually Exclusive

In a Venn diagram with 2 circles, there are 6 main parts to consider.

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Use the Venn Diagram on the right to explain how the answer to the probability question was derived.

a. P(English) = 14/40 = 7/20 Total number of students taking English out of the total number of students

b. P(Chemistry) = 29/40 ______________________________________________

c. P(only English) 9/40 ________________________________________________

d. P(only Chemistry) = 24/40 = 3/10 _____________________________________

e. P(English and Chemistry) = 5/40 = 1/8 _________________________________

f. P(English or Chemistry) = 38/40 = 19/20 _______________________________

g. P(Not English nor Chemistry) = 2/40 = 1/10 _____________________________

h. P(English|Chemistry) = 5/29 _________________________________________

i. P(Chemistry|English) = 5/14 _________________________________________

Now you try some.

1) A survey of couples in a city found the following probabilities:

• The probability that the husband is employed is 0.85.

• The probability that the wife is employed is 0.60.

• The probability that both are employed is 0.55.

A couple is selected at random. Find the probability that

a. at least one of them is employed.

b. neither is employed.

c. that only the husband is employed

d. that the wife is employed but the husband is not

e. that the wife is employed when the husband is already employed

2) In a group of 35 children, 10 have blonde hair, 14 have brown eyes, and 4 have both blonde hair and brown eyes. If a child is selected at random, find the probability that:

a. the child has blonde hair or brown eyes.

b. the child has only blonde hair

c. the child does not have brown eyes.

d. the child has neither brown eyes nor blonde hair.

e. the child has brown eyes given that the child has blonde hair.

3) Of 400 college students, 120 are enrolled in math, 220 are enrolled in English, and 55 are enrolled in both. If a student is selected at random, find:

a. P(the student is enrolled in mathematics)

b. P(the student is enrolled in mathematics or English)

c. P(the student is enrolled in either mathematics or English, but not both)

d. P(a student is enrolled in math|they are enrolled in English)

e. P( a student is enrolled in English| they are enrolled in math)

f. P(the student is enrolled in neither)

g. P(the student is enrolled in only mathematics)

h. P(the student is enrolled in only English)

i. P(the student is enrolled in English and mathematics)

Determining Outcomes and Basic Probability

1. Suppose you select a number at random from the set {90, 91, 92, …, 99} Event A is selecting multiple of 3. Event B is selecting a multiple of 4.

List the outcomes that are in:

a. Event A and Event B

b. Event A or Event B

c. Not in Event A nor in Event B

d. Event A

e. Event B

f. Event B given A

g. Event A given B

2. The following data was collected to see what people in the US do with their trash.

Municipal Waste Collected in the U.S. (millions of tons)

|Material |Recycled |Not Recycled |Totals |

|Paper |34.9 |48.9 |83.8 |

|Metal |6.5 |10.1 |16.6 |

|Glass |2.9 |9.1 |12 |

|Plastic |1.1 |20.4 |21.5 |

|Other |15.3 |67.8 |83.1 |

|Totals |60.7 |156.3 |217 |

a. Find the probability that a sample of recycled waste was paper.

b. P(plastic|not recycled)

c. P(recycle|metal)

3. Researchers asked shampoo users whether they apply shampoo directly to the head, or indirectly using a hand.

| |Directly onto head |Into hand first |Totals |

|Male |2 |18 |20 |

|Female |6 |24 |30 |

|Totals |8 |42 |50 |

a. Find the probability that a respondent applies shampoo directly to the head, given that the respondent is female

b. Find the probability that a respondent applies shampoo into their hand first.

c. Find the probability that a respondent is a male given that that respondent applies shampoo into their hand first.

4. Examine the table below. It describes the type of weather for the 30 days in April 2010.

|1. |2. |3. |4. |5. |6. |

|Rain |Cloudy |Sunny |Cloudy |Sunny |Sunny |

|7. |8. |9. |10. |11. |12. |

|Rain |Rain |Cloudy |Cloudy |Cloudy |Sunny |

|13. |14. |15. |16. |17. |18. |

|Sunny |Rain |Rain |Rain |Sunny and Very Windy |Sunny |

|19. |20. |21. |22. |23. |24. |

|Cloudy |Rain and Very Windy |Rain |Cloudy |Sunny |Sunny |

|25. |26. |27. |28. |29. |30. |

|Sunny |Sunny |Cloudy |Cloudy and Very Windy|Rain |Sunny |

a. What is the probability that a day during the month of April is a sunny day? _________

b. What is the probability that a day during the month of April is a cloudy day?_________

c. What is the probability that a day during the month of April is a rainy day?___________

d. What is the probability that a day during the month of April is a windy day?__________

e. A baseball game can be played on any day as long as it does not rain. If a day is chosen at random during the month of April to play a game of baseball, what is the probability that the athletes will be able to play?

Geometric Probability

1. A dart is thrown at random onto a board that has the shape of a circle as shown below. Calculate the probability that the dart will hit the shaded region.

2. In the figure below, PQRS is a rectangle, and A, B, C, D are the midpoints of the respective sides as shown. An arrow is shot at random onto the rectangle PQRS. Hint** Label each segment of the rectangle as a variable or number. Remember that some of the segments are congruent.

Calculate the probability that the arrow strikes:

a. triangle AQB

b. a shaded region

c. either triangle BRC or the unshaded region.

3. A game at the state fair has a circular target with a radius of 10.7 cm on a square board measuring 30 cm on a side. Players win prizes if they throw a dart and hit the circular area only.

a. What is the probability of a player winning?

b. If 25 players were to play this game, how many would you expect to win?

4. A sky‐diver jumped out of an airplane and headed to land in a trapezoid shaped grassland. When measured from above, the height of the trapezoid was 30m and the length of one base was 70m and that of the other was 40m. The grass land also contained a circular pond with a radius of 10m.

a. What is the probability that the skydiver will land in the pond?

b. What is the probability that he will land in the grassland?

5. Consider the diagram.

a. What is the total area of the square?

b. What is the area of the shaded region?

c. What is the probability that any randomly chosen point within the square will be in the shaded area?

d. What is the probability that the randomly chosen point will not land in the shaded area?

e. If 540 points were plotted, how many points would be expected to land in the shaded region?

Tree Diagrams and Compound Events

The goal of a tree diagram is to list all the different outcomes of compound events. It is an easier way of displaying the information than listing each and every outcome. Look at the following example, and then answer the questions about the diagram.

Suppose that an urn contains 2 blue balls, 1 red ball, and 4 green balls.

Two balls are chosen, with replacement, from the urn.

Assume that all balls are equally likely to be chosen.

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Now consider this example. Then answer the questions.

Suppose that an urn contains 2 blue balls, 1 red ball, and 4 green balls.

Two balls are chosen, without replacement, from the urn.

[pic]

It is now your turn to answer a problem.

1. A student in Buffalo New York made the observations below.

• Of all snowfalls, ,5% are heavy (at least 6 in)

• After a heavy snowfall, schools are closed 67% of the time.

• After a light (less than 6 in) snowfall, schools are closed 3% of the time.

Find the probability that:

a. The snowfall is light and the schools are open.

b. The snowfall is heavy and the schools are open

c. The snowfall is light and the schools are closed

d. The snowfall is heavy and the schools are closed

e. If there is any snow at all the schools are open.

f. If there is any snow at all the schools are closed

2. A bag contains three red blocks, six blue blocks, and two yellow blocks. Suppose you draw a block at random, do not return it to the bag, and then draw another block

Find the probabilities of

a. P(red and red) b. P(red and then blue)

c. P(red and blue) d. P(1 yellow)

e. P(at least 1 yellow) f. P(not yellow)

Arrangements

Find the number of arrangements for each situation.

1. Three musicians will be chosen out of a 12 member band.

2. Choosing a committee of 5 from the members of a class of 15.

3. Choosing 2 co-captains of the basketball team on a team of 12

4. Choosing the placement of 9 model cars at a dealership

5. Choosing 3 deserts from a desert tray of 8

6. Choosing a chairperson and an assistant chairperson for a committee out of 10 people.

7. Choosing 4 paintings to be displayed at 4 different locations

8. Choosing 2 paintings to be displayed at 4 different locations

9. Choosing a line up for the 9 batters batting on a baseball team of 9.

10. Choosing 4 people to run in a race out of 9 runners.

11. Given three no homework passes out to a class of 20.

12. Given three passes out to students. One is for one day no homework, another is for two days no homework, and the other one is no homework for life out of a class of 20.

13. You have 4 shirts, 5 pants, 4 hats and 2 shoes. If you will wear one of each item, how many possible combinations are there?

14. How many possible 4 digit numbers can be made if the thousands place is a 2 and units place is either a 4 or a 7.

15. You are making a five digit code. It can consist of letters and digits. How many possibilities are there?

16. You are making a four digit code each. It can consist of letters and digits but they cannot repeat. How many possibilities are there?

17. You are in a class of 20 and four people are going on a class trip. How many possible groups of 4 can be made?

18. There are 100 boys in the school. One girl had eight valentines. How many different possibilities are there?

19. You are in a class of 20. One student is winning a free trip to Great Adventure, another student is winning a no homework pass and one student is winning a marker. How many different ways are possible?

20. If I arrange 5 colored pencils, how many orders are possible?

21. Ms. Masi has 12 books that she really enjoys reading. During the summer, she is planning on assigning 3 of these books as summer reading for next year’s AP English class. How many different combinations of books could she assign her students to read?

22. Four friends want to join the “Wave of Hope” 3-on-3 basketball tournament being held next week. If only three players can be on the court at any given time, how many different teams could their team create?

23. Ten junior candidates are running for class office at Long Branch High School for next school year. There are President, Vice President, Secretary, and Treasurer offices to fill. In how many different ways could these ten candidates be elected into these offices?

24. The last summer Olympics were held in September 2008. If 9 people qualify for the world Olympic championship race and are all equally likely to win a medal, how many ways could the Olympic medals be awarded?

Answers

Arrangements

1. 220 2. 3003 3. 66 4. 56 5. 56 6. 5040

7. 24 8. 12 9. 362,880 10. 126 11. 1140 12. 6,840

13. 160 14. 200 15. 60,466,176 16. 1,413,720

17. 4,845 18. 186,087,894,300 19. 6,840 20. 120 21. 7,920

22. 4 23. 5,040 24. 504

Tree Diagrams and Compound Events

a. 9; no b. Independent c. 2/49 d. 4/49 e. 34/49

a. Without replacement b. Blue given blue; Green given blue

c. Dependent d. 1/21 e. 2/21 f. 15/21

1. a. about 92% b. about 2% c. about 3% d. about 3%

e. about 94% f. about 6%

2. a. 3/55 b. 9/55 c. 18/55 d. 18/55 e. 19/55

f. 36/55

Venn Diagrams and Mutually Exclusive Events

a. Total number of chemistry out of the total number of students

b. Number of only English out of the total number of students

c. Number of only Chemistry out of the total number of students

d. Number of students taking both out of the total number of students

e. Number of students taking only English + Number of students taking only Chemistry + Number of students taking both out of the total number of students

f. Number of students taking neither out of the total number of students

g. Number of students taking both out of the total Chemistry

h. Number of students taking both out of the total English

1. a. 90% b. 10% c. 30% d. 5% e. about 65%

2. a. About 57% b. About 17% c. 60% d. About 43% e. 40%

3. a. 30% b. About 71% c. About 58% d. 25% e. About 46%

f. About 29% g. About 16% h. About 41% i. About 14%

Determining Outcomes and Basic Probability

1. a. 96 b. 90, 92, 93, 96, 99 c. 91, 94, 95, 97, 98 d. 90, 93, 96, 99

e. 92, 96 f. 92 g. 92

2. a. About 57% b. About 13% c. About 39%

3. a. 20% b. 84% c. About 43%

4. a. 40% b. 30% c. 30% d. 10% e. 70%

Geometric Probability

1. 75% 2. a. About 13% b. 50% c. About 63% 3. a. About 40%

b. About 10 players 4. a. About 19% b. About 81% 5. a. 144 square units

b. 44 square units c. About 31% d. About 69% e. About 167 points

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a. How many different possibilities are there of choosing 2 balls? Does the diagram consider Blue and Red the same as Red and Blue?

b. Are the events independent or dependent?

c. What is the probability that you would choose a Blue and then a Red?

d. What is the probability you would choose a blue and red where order is not important?

e. What is the probability of not choosing two greens?

a. What is the difference between the events in this example and the last one?

b. Why is P(B|B) = 1/6? P(G|B) = 4/6?

c. Are these events independent or dependent? How can you tell by the diagram? How can you tell by reading the problem?

d. What is the probability that you would choose a Blue and then a Red?

e. What is the probability you would choose a blue and red where order is not important?

f. What is the probability of not choosing two greens?

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