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Geometry – Chapter 14 Day #5

Topic: Probability Distributions, Frequency Tables, Two Way Tables, Conditional Probabilities

Standards/Goals:
ü  S.CP.4.: I can use a two-way table involving categories to determine probabilities.
ü  S.CP.5.: I can recognize the concepts of conditional probabilities and independence in everyday situations.
ü  S.CP.6.: I can calculate a conditional probability and interpret the result in the context of the given problem.

We first want to look at another situation the requires the use of a conditional probability:

EXAMPLE: The table below shows the numbers of participants in an academic competition. Use this information for exercises below:
ACADEMIC COMPETITION
MALE / FEMALE / TOTALS
Freshman / 3 / 5 / 8
Sophomores / 6 / 4 / 10
Juniors / 7 / 5 / 12
Seniors / 4 / 6 / 10
TOTALS / 20 / 20 / 40
#1. What is P(female)? #2. What is P(freshman)?
#3. What is P(female freshman)? #4. What is P(female ⎸ freshman)?
#5. What is P(freshman ⎸ female)? #6. What is P(male ⎸senior)?
#7. What is P(sophomore ⎸male)? #8. What is P(junior⎸female)?

EXAMPLE: The table below shows the number of people who own road bikes or mountain bikes, and whether or not they ride their bikes to work.

Rides to work / Does not ride to work / TOTALS
Mountain Bike / 12 / 58 / 70
Road Bike / 18 / 12 / 30
TOTALS / 30 / 70 / 100

What is the probability that a randomly chosen person owns a road bike AND rides is to work?

ü  This type of table is called a two-way frequency table, used to display the frequencies of data in two different categories.

ü  To find the probability, calculate the relative frequency:

The probability that a randomly chosen person owns a road bike and rides it to work is ______.

The table below shows the amount of sleep for workers on the night shift and day shift:

Sleeps less than 8 hours / Sleeps 8 or more hours / TOTALS
Night shift / 12 / 58 / 70
Day shift / 14 / 16 / 30
TOTALS / 26 / 74 / 100

#1. What is P(sleeps less than 8 hours and works on the night shift)?

#2. What is P(sleeps 8 or more hours and works on the day shift)?

#3. What is P(sleeps 8 or more hours and works on the night shift)?

CONDITIONAL PROBABILITIES:

ü  What is the likelihood that you will pass your next test? Will your chance be better if you study more? This type of situation is called conditional probability, where probability is affected by another event, which in this case is studying.

Conditional Probability Formula
For any two events A and B, the probability of B occurring, given that event A has occurred, is:
PB⎸A=P(A and B)P(A), where P(A) ≠ 0.

EXAMPLE: The table below shows the number of students who passed and failed the last math test and the amount of time that they studied and prepared. What is the probability that a randomly chosen person failed given that he studied 4 or more hours?

ü  The term ‘given’ tells you that the category you are concerned with is an event that has already happened. Of all the students, 36 studied 4 or more hours. Of those students, 2 failed.

ü  It is written as: P(failed ⎸studied 4 or more hours) and is calculated as:

ü  The probability that a randomly chosen student failed given that he or she studied 4 or more hours is:

PASS / FAIL / TOTALS
Studied less than 4 hours / 8 / 6 / 14
Studied 4 or more hours / 34 / 2 / 36
TOTALS / 42 / 8 / 50

EXAMPLE: The table below shows the number of cars and trucks that are red or blue at a local dealership. Use the table to determine the probabilities.

RED / BLUE / TOTALS
CARS / 4 / 10 / 14
TRUCKS / 4 / 6 / 10
TOTALS / 8 / 16 / 24

#1. What is P(red ⎸ truck)? #2. What is P(blue⎸truck)?

#3. What is P(blue⎸car)? #4. What is P(red⎸car)?

EXAMPLE: Pharmaceutical Testing: In a study designed to test the effectiveness of a new drug, half of the volunteers received the drug. The other half of the volunteers received a placebo, (which is a tablet or pill containing no medication). The probability of a volunteer receiving the drug and getting well was 45%. What is the probability of someone getting well, given that he receives the drug?

EXAMPLE: Pets: In a survey of pet owners, 45% own a dog, 27% own a cat, and 12% own both a dog and a cat.

  1. What is the conditional probability that a dog owner also owns a cat?
  1. What is the conditional probability that a cat owner also owns a dog?
  1. The same survey showed that 5% of the pet owners own a dog, a cat and at least one other type of pet. What is the conditional probability that a pet owner owns a cat and some other type of pet, given that they own a dog?

HOMEWORK – Chapter 14 Day #5

Use the two-way frequency table below. It shows the number of one doctor’s female patients who caught a cold one winter and whether or not they exercised regularly.

Caught a cold / Did NOT catch a cold / TOTALS
Exercised / 8 / 30 / 38
Did NOT exercise / 10 / 2 / 12
TOTALS / 18 / 32 / 50

#1. How many patients exercised?

#2. What is the probability that a randomly chosen patient caught a cold and did not exercise?

#3. What is the probability that a randomly chosen patient exercised and did not catch a cold?

#4. What is P(did not exercise ⎸ did not catch a cold)?

The table below shows the students in a physical education class. Use this information for the exercises below:

Has played tennis / Has NOT played tennis / TOTALS
BOYS / 10 / 6 / 16
GIRLS / 10 / 4 / 14
TOTALS / 20 / 10 / 30

#5. What is P(girl)? #6. What is P(boy)?

#7. What is P(has NOT played tennis)? #8. What is P(has played tennis)?

#9. What is the probability that a randomly chosen student has played tennis given that he is a boy?

#10. What is the probability that a randomly chosen student has NOT played tennis given that she is a girl?

Use the table below. It shows the relative frequencies of students in a science club who have pets, and whether or not they have a yard.

PETS / NO PETS / TOTALS
Yard / 0.60 / 0.05 / 0.65
No yard / 0.25 / 0.10 / 0.35
TOTALS / 0.85 / 0.15 / 1

#11. What is the probability that a randomly chosen student has a yard given that they have pets?

#12. What is P(does NOT have a yard ⎸has no pets)?

#13. What is P(has no yard)? #14. What is P(has pets)?

#15. What is P(has no pets ⎸has yard)? #16. What is P(has no yard⎸ has pets)?

#17. (Use the pets example from the previous page):

ERROR ANALYSIS: Your friend determines that P(has a yard ⎸has no pets) is 0.08. What error did your friend make? What is the correct probability?

A biologist surveyed one type of plant growing on a wooded acre. Use his results, shown in the table below, for the following two exercises:

Lobed Leaves / Non-lobed Leaves / TOTALS
Red Berries / 12 / 48 / 60
No Red Berries / 40 / 0 / 40
TOTALS / 52 / 48 / 100

#18. What is P(has red berries ⎸ has lobed leaves)?

#19. What is P(has lobed leaves ⎸ has red berries)?

#20. ALLOWANCE: Suppose that 62% of children are given a weekly allowance, and 38% of children do household chores to earn an allowance. What is the probability that a child does household chores, given that the child gets an allowance?

#21. SOFTBALL: Suppose that your softball team has a 75% chance of making the playoffs. Your cross-town rivals have an 80% chance of making the playoffs. Teams that make the playoffs have a 25% chance of making the finals. Use this information to find the following probabilities:

  1. P(your team makes the playoffs and the finals)
  1. P(cross-town rivals make the playoffs and the finals)

#22. Science (STEM): In a research study, one third of the volunteers received drug A, one third received drug B, and one third received a placebo. Out of all the volunteers, 10% received drug A and got better, 8% received drug B and got better, and 12% received the placebo and got better.

  1. What is the conditional probability of a volunteer getting better if they were given drug A?
  1. What is the conditional probability of a volunteer getting better if they were given drug B?
  1. What is the conditional probability of a volunteer getting better if they were given the placebo?