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Bell Work:
Factor x – 6x – 16
2
Answer:
(x – 8)(x + 2)
Lesson 70:
Probability and
Designated Order
The study of probability began when
people began studying games of
chance such as flipping coins, rolling
dice, drawing cards from a deck, or
drawing marbles from an urn.
Problems from games of chance still
provide the best models on which to
base a study of elementary
probability, and we will concentrate
on these problems.
The study of probability is based
on the study of outcomes that
have an equal chance of
occurring.
It is customary to call activities
such as flipping coins, rolling dice,
blindly selecting cards from a
deck, and drawing marbles from
an urn experiments and to call the
individual results outcomes.
We call the set of equally probable
outcomes the sample space for
the experiment. A toss of a fair
coin has two equally probable
outcomes. Thus, the sample
space for a coin toss is heads or
tails, as shown below.
H
T
The roll of a single die has six
equally probable outcomes. Thus,
the figure below shows the sample
space for the roll of a single die.
1
2
3
4
5
6
We define the probability of a particular
even as the number of outcomes that
satisfy the requirement divided by the
total number of outcomes in the sample
space.
particular event=number outcomes that satisfy requirement
total number of outcomes in sample space
The probability of any event is a
number between 0 and 1 inclusive.
If no outcomes satisfy the
requirement, the probability is 0,
and if every outcome satisfies the
requirement, the probability is 1.
Thus we see that a probability of 2 of 7 ½ is not possible because
the probability of any event must
be a number between 0 and 1.
Example:
A fair coin is tossed three times
and comes up heads every time.
What is the probability that on the
next toss it will come up heads?
Answer:
P=
number of outcomes
outcomes in sample space
P=½
Example:
Six green marbles and eight red
marbles are placed in an urn. One
marble is drawn and then dropped
back in the urn. Then a second
marble is drawn and dropped back
into the urn. Both marbles were red.
If another marble is drawn, what is
the probability that it will be red?
Answer:
P = 8/14 = 4/7
Practice:
A single die is rolled three times.
The results are 1, 4, and 3, in that
order. What is the probability that
the next roll will produce a number
greater than 2?
Answer:
4/6 = 2/3
Practice:
Two dice are rolled. What is the
probability that the sum of the
numbers rolled is
a) 7
b) A number greater than 8
Answer:
a) 6/36 = 1/6
b) 10/36 = 5/18
Designated Order:
The probability of future outcomes
of independent events happening
in a designated order is the
product of the probability of the
individual outcomes.
For example, if we toss a coin twice,
the probability of getting a heads on
the first toss and a tails on the
second toss is one fourth.
P(H, T) = P(H) x P(T) = ½ x ½ = ¼
Example:
A fair coin is tossed four times.
What is the probability that the first
two times it comes up heads and
the last two times it comes up
tails?
Answer:
½ x ½ x ½ x ½ = 1/16
Practice:
The spinner show is spun twice.
What is the probability that the
spinner stops on 4 and then on 3?
Answer:
¼ x ¼ = 1/16
HW: Lesson 70 #1-30