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AP Statistics: Chapter 14 Reading Summary
Chapter 14: From Randomness to Probability
In Statistics, what is a random phenomenon?
 We call phenomenon random if individual outcomes are uncertain but there in nonetheless a
regular distribution of outcomes in a large number of repetitions. “Random” in statistics is not
a synonym for “haphazard” but a description of a kind of order that emerges only in the long
run.
Define the following:
 Probability
 The probability of any outcome of a random phenomenon is the proportion of times the
outcome would occur in a very long series of repetitions. That is, probability is a long-term
relative frequency.
number of outcomes corresponding to event E
 The probability of an event E, P  E  
total number of outcomes
 Trial
 Each occasion in which we observe a random phenomenon (ex. Roll of dice)
 One of the repetitions of a statistical experiment having exactly two mutually exclusive
outcomes each with a constant probability of occurrence.
 Trials are independent if the outcome of one trial does not influence the outcome of any
other.
 Outcome
 An outcome is the result of an experiment or other situation involving uncertainty.
 A value obtained for a trial (what actually happened or could happen)
 Event
 An event is any outcome or a set of outcomes of a random phenomenon. That is, an event
is subset of the sample space.
 Probability Model
 A mathematical description of a random phenomenon consisting of two parts: a sample
space (S) and a way of assigning probabilities to events.
What does it mean for two events to be “independent events”?
 Two events are said to be independent if the outcome of one of them does not influence the
other. For example, in sporting events, the outcomes of different games are usually
considered independent even though that may not be true in a completely strict and literal
sense.
 The multiplication rule for independent events says that if A and B are independent, then
P(A and B) = P(A) × P(B).
 Two events A and B are independent if P(B|A) = P(B) .
What is the Law of Large Numbers?
 In statistical terms, a rule that assumes that as the number of samples increases, the
average of these samples is likely to reach the mean of the whole population, i.e., as the
number of independent trials increases, the probability based on our observations will get
closer and close to the theoretical probability of the event.
RNBriones
Concord High
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AP Statistics: Chapter 14 Reading Summary
What does a probability of 0 mean? A probability of 1?
 An event with probability 0 never occurs. An event with probability 1 occurs on every
trial. The probability of any event must be between these two possible outcomes, i.e.,
0  P E 1
What is the sum of all possible outcomes in a probability experiment? What is this called?
 Because some outcome must occur on every trial, the sum of the probabilities for all possible
outcomes must be exactly 1.
 It is called the Sample Space.
Examples:
1) Experiment Rolling a die once:
Sample space S = {1, 2, 3, 4, 5, 6}
2) Experiment Tossing a coin:
Sample space S = {Heads, Tails}
What is meant by the union of two or more events?
 The union of any collection of events is the event that at least one of the collections occurs.
What is the Complement Rule?
 The complement of any event A is the event that A does not occur, written as Ac .
Complements are “opposites” (ex. “rain” or “not rain”)

 
The complement rule states that P Ac  1  P ( A) ., i.e., P(not A) = 1 – P( A)
What does it mean to be disjoint? What is another term sometimes used for this?
 Two events A and B are disjoint (also called mutually exclusive) if they have no outcomes in
common (no intersection) and so can never occur simultaneously.
What is the Addition Rule for disjoint events?
 If events A, B, and C are disjoint in the sense that no two have any outcomes in common,
then P(one or more of A, B, C) = P(A) + P(B) + P(C).
General Addition Rule for unions of two events
 For any two events A and B,
P(A or B) = P(A) + P(B) – P(A and B).
What is the difference between sampling with replacement and sampling without replacement?
 For example, if you are selecting random digits by drawing numbered slips of paper from a
hat, and you want all ten digits to be equally likely to be selected each draw, then after you
draw a digit and record it, you must put it back into the hat. Then the second draw will be
exactly like the first. This is referred to as sampling with replacement. If you do not replace
the slips you draw, however, there are only nine choices for the second slip picked, and eight
for the third. This is called sampling without replacement.
RNBriones
Concord High
Name:
Date:
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AP Statistics: Chapter 14 Reading Summary
What is the Multiplication Rule for independent events?
 Two events A and B are independent if knowing that one occurs does not change the
probability that the other occurs.
 If A and B are independent, then P(A and B) = P(A) × P(B).
Can disjoint events be independent events? Explain.
 No, disjoint events cannot be independent. If A and B are disjoint, then the fact
that A occurs tells us that B cannot occur.
 If we know they are disjoint, then if event A happened, we know that event B did not
happen, therefore, knowledge of event A affected our knowledge of event B, making them
dependent events.
Ex) Suppose A and B are disjoint; then P(A and B) = 0.
Suppose A and B are independent; then P(A and B) = P(A) × P(B).
Hence P(A) × P(B) = 0 so that P(A) = 0 or P(B) = 0.
This is impossible if both events have some chance of occurring.
Ex) Consider two complementary events. They are clearly disjoint; are they independent? The
answer is (almost always) NO: if event A occurs, then event B CANNOT occur, so even
intuitively they "affect" each other, and so are not independent.
General Probability Rules
What is meant by joint probability?
 The simultaneous occurrence of two events is called a joint event. The probability of a
joint event is called a joint probability.
What is meant by conditional probability?
 The conditional probability, P(B|A), is the probability that event B occurs given that
event A has already occurred.
State the General Multiplication Rule.
 The probability that both of two events A and B happen together can be found by
P(A and B) = P(A) × P(B|A).
How is the general multiplication rule different than the multiplication rule for independent events?
 If A and B are independent, then P(B|A) = P(B) so the two rules are the same.
If A and B are not independent, the rule must be adjusted for this condition.
State the formula for finding conditional probability.

When P(A) > 0, the conditional probability of B given A is:
P  B | A 
P ( A and B )
.
P ( A)
What is meant by the intersection of two or more events?
 The intersection of any collection of events is the event that all of the events occur.
Explain the difference between the union and the intersection of two or more events.
 The union is the event that any of them occur, the intersection is the event that
they all occur.
State the formula used to determine if two events are independent.
 Two events A and B are independent if P(B|A) = P(B) .
RNBriones
Concord High
Name:
Date:
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AP Statistics: Chapter 14 Reading Summary
Warning:
Don't confuse the notions of “independent events” and “disjoint events”. Independence
means that the outcome of one event does not influence the outcome of the other. Disjoint
means that if one event occurs then the other cannot occur—the very opposite of
independence!
Practice:
1) Are “Rolling a 3 and a 4 on one roll of a die” mutually exclusive events?
2) Are “Rolling a 3 on one die and rolling a 4 on another” mutually exclusive events?
3) Are “One student being both a boy and tall” mutually exclusive events?
4) Are “Rolling a die twice and getting 3 on one die and a 4 on the other” independent events?
5) Are “Pulling an ace from a pack of cards, not returning it to the deck, and pulling another ace”
independent events?
6) Are “Pulling an ace from a pack of cards, returning it to the deck, and pulling another ace”
independent events?
7) P(Having Brown Hair) = 0.30, and P(Having Red Hair) = 0.10.
8) If P(Having Brown Hair and Having Red Hair) = 0.20, are the two events independent?
RNBriones
Concord High