Course:FRST 231 DE/Rules of Probability

This quiz is designed to help you gain confidence with the concepts covered in Module 2 of FRST 231. Do your best to solve each question, then click "Submit" for the answers and an explanation.

Topics covered in this quiz include:

• Rules of probability
• Conditional probability
• Venn diagrams
• Probability trees
• Bayesian probability
• Counting techniques

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NOTE: Unless the question states otherwise, please write all probabilities as decimals (i.e. numbers between 0 and 1) instead of percentages (i.e. a number between 0 and 100).

Question 1

Which of the following are rules of probability?

	$P(A\cap B)=P(A)*P(B)$
	$P(A\cap B)=P(A)*P(B|A)$
	$P(A\cup B)=P(A)+P(B)$
	$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
	$P({\bar {A}})=1-P(A)$
	$P({\bar {A}})=P(A)-1$, where $P(A)<1$

Question 2

A gardener is pulling seeds out of a bag containing 5 pumpkin seeds, 4 sunflower seeds, and 3 watermelon seeds. Assuming that the type of seed doesn't affect its chance of being selected, what are the following probabilities if the gardener pulls out 2 seeds (one after the other)? (NOTE: Round to 2 decimal places.)

Selecting a watermelon seed on the first draw

Selecting a watermelon seed OR a sunflower seed on the first draw

NOT selecting a watermelon seed on the first draw

Selecting 2 sunflower seeds with replacement

Selecting 2 sunflower seeds without replacement

Selecting 1 sunflower and 1 pumpkin seed without replacement

Selecting 1 sunflower and 1 pumpkin seed with replacement

Mountain pine beetle infestation

Question 3

Within a lodgepole pine (Pinus contorta var. latifolia) forest, 40% of the trees are infested with mountain pine beetle, 15% have dwarf mistletoe, and 5% have both mountain pine beetle and dwarf mistletoe. Calculate the following probabilities. (Answer to 2 decimal places)

Selecting a tree with both mountain pine beetle and dwarf mistletoe

Selecting a non-infected tree

Selecting a tree with either mountain pine beetle or dwarf mistletoe, but not both

Question 4

For Question 2 above, which of the following Venn diagrams is correct?

	A
	B
	C

The face cards: jack, queen, king.

Question 5

The average deck of cards has four suits. Within each suit there are 3 face cards (jack, queen, king) and 10 non-face cards (ace, 2 to 10).

Complete the following tree diagram which shows the conditional probabilities of receiving face cards or non-face cards when three cards are randomly drawn without replacement. A few answers are provided to get you started. NOTE: It will be easier to answer the following questions if you draw your own tree diagram and fill your answers in. (Round answers to 3 decimal places)

A

B

C

D

E

F

G

H

I

J

K

Question 6

Using the tree diagram from the previous question, calculate the probability for each 3-card combination, given below as $P(draw1\cap draw2\cap draw3)$. NOTE: Add these answers to the tree diagram that you drew for the previous question to answer the next questions. (Round answers to 3 decimals)

$P(face\cap face\cap face)$

$P(face\cap face\cap non\!-\!face)$

$P(face\cap non\!-\!face\cap face)$

$P(face\cap non\!-\!face\cap non\!-\!face)$

$P(non\!-\!face\cap face\cap face)$

$P(non\!-\!face\cap face\cap non\!-\!face)$

$P(non\!-\!face\cap non\!-\!face\cap face)$

$P(non\!-\!face\cap non\!-\!face\cap non\!-\!face)$

Question 7

Use your tree diagram with the answers from questions 5 and 6 to help find the following probabilities for drawing 3 cards.

Drawing at least 1 face card

Drawing 1 or 3 face cards

Question 8

The table below contains enrolement data for a number of faculties at UBC's Vancouver campus in 2021.

	Forestry	Applied science	Arts	Science	LFS	Business	Total
Undergraduates	1179	5712	14861	9605	1833	5767	38957
Graduate students	374	2350	1966	1758	239	794	7481
Total	1553	8062	16827	11363	2072	6561	46438

Based on this data, what are the following probabilities if you select a student at random? (Round to 3 decimal places)

Selecting an undergraduate in Forestry

Given that they are in Forestry, selecting an undergraduate

Given that they are an undergraduate, selecting a forestry student

Given that they are a grad student, selecting a Science or LFS student

Given that they are a science or LFS student, selecting a grad student

SARS-CoV-2 virus which causes the COVID-19 disease.

Question 9

A recent study looked at 422,966 people in Los Angeles who caught COVID-19. Within this sample, 141,928 were unvaccinated, 224,853 had received two vaccines, and 56,185 had received a booster (3 total vaccines). Within the unvaccinated group, 3,989 people were hospitalized due to COVID, while 2,295 people were hospitalized from the group with two vaccines and 413 people were vaccinated from the group with the booster.

What is the overall probability (to 3 decimals) of being hospitalized due to COVID-19?

A person arrives in a Los Angeles hospital with COVID. What is the probability (rounded to 3 decimals) that they...

...are unvaccinated?

...are double-vaccinated?

...have received a booster shot?

Question 10

Match each scenario with the most appropriate counting method.

A. Finding the number of possible hands in a game of poker

B. Finding the number of possible outcomes from flipping a coin 100 times

C. Finding the number of possible ways of arranging potted plants in a window display

Simple exponentiation ($n^{k}$)

Combination ($C{\binom {n}{r}}$)

Permutation ($P{\binom {n}{r}}$)

Question 11

12 UBC students are at the Bookstore trying to purchase their course materials.

If all 12 students form a single line at the cash register, how many ways unique ways can they arrange themselves?

The group decides that 6 students will wait in a single line while 6 students will hang out at Starbucks. How many possible combinations of line-standers and coffee-sitters are there?

The 6 students standing in line partner up with each other. Each pair of students stays together, but they can vary which student is standing in front of the other. How many ways are there for these 6 students to line up now?

Question 12

A keypad uses the numbers 0-9 to make passwords which are 4 digits long.

How many potential passwords can be be made?