Transcript 11-2 Basic Probability

11-2 Basic Probability
Theoretical vs. Experimental
◆ Theoretical
probabilities are those that
can be determined purely on formal or
logical grounds, independent of prior
experience.
◆ Experimental probabilities are estimates
of the relative frequency of an event
based
by
our
past
observational
experience.
Experimental Probability
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Example 1—Experimental
Probability
◆ Out
of the 60 vehicles in the teacher parking lot
today, 15 are pickup trucks. What is the probability
that a vehicle in the lot is a pickup truck?
◆ GOT
IT? A softball player got a hit in 20 of her last 50
times at bat. What is the probability that she will get a
hit in her next at bat?
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Probability Experiment
◆ Can
be repeated many times
(at least in theory)
◆ Each such repetition is called a trial
◆ When an experiment is performed it
can result in one or more outcomes,
which are called events.
Law of Large Numbers
◆ The
more repetitions we take, the more closely
the experimental probability will reflect the true
theoretical probability.
◆ This
is sometimes referred to as the Law of Large
Numbers, which states that if an experiment is
repeated a large number of times, the relative
frequency of the outcome will tend to be close to
the theoretical probability of the outcome.
Summary of 20,000 Coin Tosses
Num of Tosses
n
10
100
1,000
5,000
10,000
15,000
20,000
Num. of Heads
f
8
62
473
2,550
5,098
7,649
10,038
Relative Freq.
f/n
.8000
.6200
.4730
.5100
.5098
.5099
.5019
Example 2—A Simulation
◆ On
a multiple choice test, each item
has four choices, but only one
choice is correct. What is the
probability that we will pass the
test by guessing? (6 out of 10 OR
60%)
◆ How can we simulate guessing? How
many trials should we run?
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Choose the numbers from 1-4, let 1
represent the correct solution and use
Using a Random # generator.
P(passing) = 1/16
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Theoretical Probability
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Example 3—Finding
Theoretical Probability
◆
What is the probability of getting a 3 on one
roll of a standard number cube?
P(3) = 1/6
◆
What is the probability of getting a sum of 4
on one roll of two standard number cubes?
P(sum 4) = 3/36= 1/12
◆
What is the probability of getting a sum that
is an odd number on one roll of two standard
number cubes?
P(sum is odd) = 1/2
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Geometric Probability
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Solution—Use Area
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Your Turn—Geometric
Probability
◆A
carnival game consists of throwing
darts at a circular board as shown.
What is the geometric probability
that a dart thrown at random will
hit the shaded circle?
ANSWER: About 14%
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11-3 Probability of Multiple
Events
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Outcomes of Different Events
◆ When
the outcome of one event affects the
outcome of a second event, we say that
the events are dependent.
◆ When one outcome of one event does not
affect a second event, we say that the
events are independent.
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Decide if the following are dependent or
independent
◆ An
expo marker is picked at
random from a box and then
replaced. A second marker is then
grabbed at random.
◆ Two dice are rolled at the same
time.
◆ An Ace is picked from a deck of
cards. Without replacing it, a Jack
is picked from the deck.
Independent
Independent
Dependent
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How to find the Probability of an “AND” for Two
Independent Events
◆
Ex: If P(A) = ½ and P(B) = 1/3 then P(A and B) =
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Your Turn
A box contains 20 red marbles and 30 blue marbles. A second
box contains 10 white marbles and 47 black marbles. If you
choose one marble from each box without looking, what is the
probability that you get a blue marble and a black marble?
The probability that a blue and a black marble will be drawn is
47
, or 49%.
95
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Probability of an A “OR” B
Independent Events Exclusive Events
If Two events are mutually
exclusive then they can not
happen at the same time.
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Mutually Exclusive??
Hint: Is it
impossible for the
events to occur at
the same time?
A spinner has ten equal-sized sections labeled 1
to 10. Find the probability of each event.
A. P(even or multiple 0f 5)
B. P(Multiple of 3 or 4)
No!
P(A)+P(B)-P(A and B)
Yes!
P(A)+P(B)
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Work them through…
Do you know what to do??
A spinner has ten equal-sized sections labeled 1
to 10. Find the probability of each event.
A. P(even or multiple 0f 5)
B. P(Multiple of 3 or 4)
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Probability of Multiple Events
A spinner has twenty equal-size sections numbered from 1 to
20. If you spin the spinner, what is the probability that the
number you spin will be a multiple of 2 or a multiple of 3?
Are the events mutually exclusive?
No. So:
P(A) + P(B) - P(A and B)
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FSA Practice Problems of the Day!
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