Chapter 4: Probability, Randomness, and Uncertainty

Question 1

probability experiment (or trial)

Answer 1

any process with a result determined by chance

Question 2

outcome

Answer 2

each individual result that is possible for a probability experiment

Question 3

sample space

Answer 3

the set of all possible outcomes for a given probability experiment

Question 4

event

Answer 4

a subset of outcomes from the sample space

Question 5

tree diagram

Answer 5

• allows the outcomes to be organized in a systematic manner
• begins with the possible outcomes for the first stage and then branches for each additional possibility
• each of the elements of the last row in the tree diagram represents a unique outcome in the sample space

Question 6

subjective probability

Answer 6

an educated guess regarding the chance that an event will occur

Question 7

experimental (empirical) probability

Answer 7

• uses the outcomes obtained by repeatedly performing an experiment to calculate the probability
• rounding rule: when calculating probability, give the exact fraction or a decimal rounded to four decimal places; if extremely small, it is permissible to round the decimal to the first nonzero digit

Question 8

Law of Large Numbers

Answer 8

the greater the number of trials, the closer the experimental probability will be to the true probability

Question 9

classical (theoretical) probability

Answer 9

• the most precise type of probability
• can only be calculated when all possible outcomes in the sample space are known and equally likely to occur

Question 10

properties of probability

Answer 10

(when an event includes the entire sample space, the probability is 1)

Question 11

complement of an event E (E^c)

Answer 11

the set of all outcomes in the sample space that are not in E

Question 12

Complement Rule for Probability

Answer 12

P(E) + P(E^c) = 1

Question 13

factorial

Answer 13

the product of all positive integers less than or equal to a given positive integer, n (0! = 1)

n! = n(n−1)(n−2) ⋯ (2)(1)

Question 14

combination

Answer 14

a selection of objects from a group without regard to their arrangement

Question 15

permutation

Answer 15

• a selection of objects from a group where the arrangement is specific (also when “repetitions are not allowed”)
• _nP_n = n!

Question 16

special permutations

Answer 16

involve objects within a group that are identical

Question 17

key terms

Answer 17

at least, at most, greater than, less than, between, etc.

Question 18

and

Answer 18

multiply

Question 19

or

Answer 19

add

Question 20

Addition Rule for Probability

Answer 20

P(E or F) = P(E) + P(F) − P(E and F)

Question 21

Addition Rule for Probability of Mutually Exclusive Events

Answer 21

when the events cannot happen at the same time:

P(E or F) = P(E) + P(F)

Question 22

Multiplication Rule for Probability of Independent Events

Answer 22

when the outcome of one event does not influence the probability of the other:

P(E and F) = P(E) ⋅ P(F)

Question 23

multistage experiment

Answer 23

• experiment with more than one step
• drawing one card from a deck vs. drawing a card, shuffling it back in, and drawing another

Question 24

with or without replacement

Answer 24

• whether or not objects from the first stage of the experiment were placed back into consideration for a subsequent stage
• with replacement: creates independent events
  P(E and F) = P(E) ⋅ P(F)
• without replacement: creates dependent events
  example: P(queen and then king) = 4/52 ⋅ 4/51

Question 25

conditional probability, P(F | E)

Answer 25

• “the probability of F given E” for dependent events
• the probability of event F occurring given that event E occurs first
• answer is only the subsequent part of multistage experiment

Question 26

Multiplication Rule for Probability of Dependent Events

Answer 26

when the probability of one is influenced by the probability of the first:

P(E and F) = P(E) ⋅ P(F | E)

OR
P(E and F) = P(F) ⋅ P(E | F)

Question 27

Fundamental Counting Principle

Answer 27

• For a multistage experiment with n stages where the first stage has k₁ outcomes, the second stage has k₂ outcomes, and so forth, the total number of possible outcomes for the sequence of stages that make up the multistage experiment is k₁ ⋅ k₂ ⋅ k₃ ⋅ … ⋅ k_n.
• example: account “numbers” with two letters followed by three numbers
  26 ⋅ 26 ⋅ 10 ⋅ 10 ⋅ 10 = 676,000