Unit 4: Probability, Random Variables, and Probability Distributions

AP Statistics | 2024-25

What is the nature of chance behavior?

What is the nature of chance behavior?

it is unpredictable in the short run, but has a regular and predictable pattern in the long run

What does a random process do?

generate outcomes that are determined purely by chance

What is a trial?

one repetition of a random process

True or false: The probability of any outcome of a chance process is always a number between 0 and 1

true

What does probability describe?

the proportion of times the outcome would occur in a very long series of repetitions

What does the law of large numbers state?

that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches its probability

True or false: Probability allows us to make short-run predictions

false

What does the law of averages assume?

that the results of a chance process have to even out in the short run

Why is the law of averages a misguided belief?

future outcomes are not affected by past behavior; past outcomes do not influence the likelihood of individual outcomes occurring in the future

Ex / According to the Bood of Odds website, the probability that a randomly selected U.S. adult usually eats breakfast is 0.61. Explain what probability 0.61 means in this setting

If you take a very large random sample of U.S. adults, about 61% of them will say they eat breakfast

What does the sum of probabilities in a probability model always equal?

1

What is the probability when an outcome is impossible and can never occur?

0

What is the probability when an outcome is certain and will occur on every trial?

1

What is the probability when an outcome is very unlikely but will occur once in a while in a long sequence of trials?

0.001

What is simulation?

the imitation of chance behavior, based on a model that accurately reflects the situation

What is the simulation process?

• ask a question about the chance process

• describe how to use a chance device to imitate one trial (repetition) and tell what you will record at the end of each trial

• perform many trails of the simulation

• use the results of your simulation to answer the question of interest

What is the purpose of simulation?

to estimate probabilities that are difficult to calculate theoretically; to represent experimental probabilities

What is a probability model?

a description of some chance process that consists of two parts:

• a list of all possible outcomes

• a probability for each outcome

What is the sample space ‘S’ of a chance process?

the list of all possible outcomes

What do probability models allow us to find?

the probability of any collection of outcomes

What is an event?

any collection of outcomes from some chance process

What events usually designated by?

capital letters

ex / probability of rain → P(R)

What is the probability that event A occurs if all outcomes in S are equally likely?

(number of outcomes in event A) / (total number of outcomes in sample space)

What is the probability that an event does NOT occur?

one minus the probability that the event does occur

P(A^C) = 1 – P(A), A^C is the complement of event A

What is the probability of two events that have no outcomes in common?

the sum of their individual probabilities

P(A or B) = P(A) + P(B)

What does P(A ∩ B) mean?

P(A and B)

What does P(A ∪ B) mean?

P(A or B)

When are two events mutually exclusive (disjointed)?

if they have no outcomes in common and therefore can never occur together

What is the addition rule for mutually exclusive events?

P(A or B) = P(A) + P(B)

How do you show that a probability model is legitimate?

by stating that the probabilities add up to 1 and all the probabilities are between 0 and 1

What is a Venn diagram?

one or more circles surrounded by a rectangle

What does each circle in a Venn diagram represent?

an event

What does the region inside the rectangle of a Venn diagram represent?

the sample space of the chance process

When is an observed result of a simulation statistically significant?

when the probability of getting that result is less than 5%

What is conditional probability?

the probability that one event happens given that another event is already known to have happened

What does P(A|B) mean?

the probability that event A happens given that event B has happened

What does P(B|A) mean?

the probability that event B happens given that event A has happened

What is the formula for the conditional probability P(A|B)?

P(A and B)

−−−−−−−−−−

P(B)

When are two events independent?

if knowing whether or not one event has occurred does not change the probability that the other event will happen

When are events A and B independent?

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

When are events B and A independent?

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

What is the multiplication rule for independent events?

If A and B are independent events, then the probability that A and B both occur is:

P(A and B) = P(A ∩ B) = P(A) × P(B|A)

What is the difference between “mutually exclusive” and “independent”?

mutually exclusive events are completely separate and cannot occur at the same time, while independent events can happen together without influencing each other

What does a tree diagram show?

the sample space of a chance process involving multiple stages, where the probability of each outcome is shown on the corresponding branch of the tree

Which probabilities on a tree diagram are conditional?

all probabilities after the first stage are conditional

What is a very common way to lose credit on probability questions?

not showing work

When is it better to use a tree diagram than a two-way table?

when you have multiple stages to be tested