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Transcript
Chapter 7: Probability
Lesson 1: Basic Principles of
Probability
Mrs. Parziale
Vocabulary
• Experiment: a situation that has several
possible results
• Outcomes: results of an experiment
• Sample space: the set of all possible outcomes
• Event: subset of the sample space
• Relative Frequency: comes from conducting an
experiment – is your results from a simulation.
In contrast, probability uses information to
predict future events.
More Terms
• Probability of an event: a number from 0 to 1 that
measures its likelihood.
Notation:
N(E) = the number of elements in a set E
S = sample space
P(E) = the probability that E occurs
Assuming each outcome in S is equally likely
N
(
E
)
# of Outcomes In Event
Formula: P(E) =

# of Outcomes In Sample Space
S
Theorem: Basic Properties of
Probability:
• If S = sample space and P(E) be the probability
of event E, then
• (i) 0  P ( E )  1
• (ii) If E = S, then P(E) = 1
• (iii) If E = {} (empty set), then P(E) = 0.
Example 1:
Toss a fair coin 2 times. Find the probability of
getting 2 heads:
• Sample space: S = { HH, TH, HT, TT } (S = 4)
• Event: 2 heads
• N(E) = # of outcomes = 1 (HH)
P(2 heads) = N ( E )  1
S
4
Example 2:
Toss a fair coin 3 times. Find the probability of
getting exactly one head:
Sample space:
S=
Event:
• N(E) (# of outcomes) =
• P(E) =
Example 3:
Roll two die. Take the sum of the numbers
on top.
(a) Event: sum of 10
P(E) =
(b) Event: sum less than 5
P(E) =
(c) Event: sum is less than 50 P(E) =
(d) Event: sum is equal to 1
P(E) =
(e) What sum has the highest probability of occurring?
Example 4:
A researcher is studying the number of boys and girls in families with three
children. Assume that the birth of a boy or a girl is equally likely.
a. Find the probability that a family of three children has exactly one boy.
First identify the sample space.
b. Find the probability that a family of three children has exactly two girls.
Example 5:
Students were testing the theoretical probability of
getting heads or tails when tossing a coin. While
performing an experiment of 2500 tosses, heads
came up 2466 times.
a. What is the probability that the 2501st coin
tossed will be heads?
b. What is the probability that the 2501st coin
tossed will be tails?
Closure
• Probability of an event can be expressed as a
number between _______ and _______ ,
inclusive.
• If the event equals the sample space, P(E) = __
• A probability of zero means ______________
• What is the difference between Probability
and Relative Frequency?