ap stats probability

sample space

the complete set of all possible, non-overlapping outcomes of a random process or chance experiment.

• roll a die S={1,2,3,4,5,6}

event

any collection of outcomes from the sample space

• rolling a prime # E={2,3,5}

• rolling a prime # or even number E = {2,3,4,5,6}

simple event

an event consisting of exactly one outcome

• rolling a “2” E = {2}

complement

for any event E, P(E) + P(not E) = 1

independent events

two events are independent if knowing that one will occur/has occurred does not change the probability that the other occurs.

+ must be able to occur simultaneously, cannot be mutually exclusive

1. a randomly selected student is a senior - what is the probability they play soccer for PWSH?

• likely independent

2. a randomly selected student is a junior - what is the probability they will graduate this school year?

• dependent

checking for independence

1. P(A and B) = P(A) * P(B)

2. P(A|B) = P(A)

3. P(B|A) = P(B)

+ P(A|B) = P(A) = P(A|B^c)

mutually exclusive/disjoint

if events are mutually exclusive, only one can occur at a time

+ two events have no outcomes in common (ex. rolling a 2 and a 5 on the same roll)

P(A and B) = 0

probability multiplication rules

if two events are indep: P(A and B) = P(A) * P(B)

not indep, use general rule: P(A and B) = P(A) * P(B|A)

probability addition rules

if two events are mutually exclusive: P(E or F) = P(E) + P(F)

if not, use general addition rule: P(E or F) = P(E) + P(F) - P(E and F)

indep events version. : P(E or F) = P(E) + P(F) - [P(E) * P(F)]

exactly one example

light bulb defective 5% of time. you randomly pick a package of two such bulbs off the shelf of a store. what is the probability that exactly one bulb is defective?

P(exactly one) = P(D and D^c) or P(D^c and D)

= (0.95)(0.95) + (0.95)(0.05)

= (0.095)

at least one

P(at least one) = 1 - P(D^c and D^c)

same example above, but probability that at least one bulb is defective

P(at least one) = 1 - P(0.95)²

= 0.0975.

conditional probability

a probability that takes into account a given condition

P(B|A) = P(A and B) / P(A)

or = P(and)/P(given)

law of large numbers

as the number of repetitions of a chance experiment increase, the difference between the relative frequency of occurrence for an event and the true probability approaches zero

• with larger n, experimental probability approaches the theoretical probability

simulation summary and drawing conclusions only appropriate

~at least 100 trials

iqr + outlier formulas

• Interquartile Range: IQR = Q₃ - Q₁

• Lower Outlier Fence: Lower Boundary = Q₁ - (1.5 × IQR)

• Upper Outlier Fence: Upper Boundary = Q₃ + (1.5 × IQR)

SD, mean-based

lower: mean - 2s

uppwer: mean + 2s

sample variance formula

population variance formula