Chapter 4 Flashcards
An experiment
A process that, when performed, results in one and only one of many observations.
Outcomes
The observations from the experiment.
Sample space
The collection of all outcomes for an experiment is called a sample space.
Ex: tree diagrams
Event
A collection of one or more of the outcomes of an experiment.
2 types of events
- Compound event
- Simple event
Simple event
A event that includes one and only one of the final outcomes for an experiment.
Denoted by Ei
Ex: tossing 2 heads or a heads tails
Compound event
A collection of more than one outcome for an experiment.
Ex: tossing at least one head
Probability
The numerical measure of the likelihood that a specific event will occur.
2 properties of probability
- The probability of an event will always lie between 0 and 1
- The sum of probabilities of all simple events (or final outcomes) for an experiment is always 1
Classical probability
If the probability in an experiment is equally likely.
Classical probability for a simple event
P = 1/ total number of outcomes
Classical probability for a compound event
P = number of outcomes favourable to a / total number of outcomes for the experiment
Relative frequency
If the outcomes of experiments are not equally likely.
Gives an approximate probability
P = frequency of event A/ sample size
Law of large numbers
If an experiment is repeated again and again, the probability of an event obtained from the relative frequency approaches the actual (or theoretical) probability.
Subjective probability
The probability assigned to an event based on subjective judgement, experience, information, and belief.
A guess.
Marginal probability
The probability of a single event without consideration of any other event.
Conditional probability
The probability that an event will occur given that another has already occurred.
Probability of A given B = P(A| B)
Mutually exclusive events
Events that cannot occur together.
Ex: toss a coin
Independent events
2 events where one does not affect the probability or the occurrence of the other.
P (a|b) = P(A) or P (b|a) = P(B)
Dependent events
If the events occur where one event affects the probability of the occurrence of the other events.
P(a|b) does not = P(A) or P (b|a) does not = P(B)
Complementary events
Complement of A = Ac or A bar
Ac = the event that includes all the outcomes for an experiment that are not in A.
Opposite of the event A.
Intersection of events
The intersection of events A and B represents the collection of all outcomes that are common to both A and B.
A and B = A n B
Joint probability
The probability of the intersection of two events.
Joint probability of independent events
P(A n B) = P(A) x P(B)
Can be extended to more than two independent events.
Joint probability of dependent events
P(A n B) = P(A) x P(B | A)
P(A n B) = P(B) x P(A | B)
Joint probability of mutually exclusive events
It is always zero.
They can never happen at the same time.
Union of events
The union of events A and B represents the collection of all outcomes that belong to either A or B or to both A and B.
A u B = A or B
Probability of 2 mutually nonexclusive events.
P(A | B) = P(A) + P(B) - P(A and B)
Probability of 2 mutually non-exclusive events.
P(A n B) = P(A) + P(B)
Calculating total outcomes
The total number of outcomes for an event comes from multiplying the number of outcomes from each individual experiment.
Factorials
n! = n factorial
Represents the products of all the integers from n to 1.
Ex: n! = 4x3x2x1 = 24
Combinations
Give the number of ways x elements can be selected - from n elements.
Order does not matter
nCx
n = total number of elements
x = denotes the number of elements selected per selection
n! / x! (n-x)!
Permutations
The order of selection does matter.
The number of permutations is always greater than or equal to the number of combinations.
Total selections of x elements from n (different) elements in such a way that the order of selections is important.
Can be called “arangements”
nPx = n! / (n-x)!
