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In a
binomial experiment there are
two
mutually exclusive outcomes, often referred to as
"success" and "failure". If the probability of success is
p, the probability of failure is 1 - p.
Such an experiment whose
outcome is random and can be either of two possibilities,
"success" or "failure", is called a
Bernoulli trial, after Swiss mathematician Jacob
Bernoulli (1654 - 1705). |
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Examples of Bernoulli trials:
• flipping a coin -- heads is success, tails is
failure
• rolling a die -- 3 is success, anything else is
failure
• voting -- votes for candidate A is success,
anything else is failure
• determining eye color -- green eyes is success,
anything else is failure
• spraying crops -- the insects are killed is
success, anything else is failure
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When computing a
binomial probability,
it is necessary to calculate and multiply three separate
factors:
1. the number of ways to
select exactly r successes,
2. the probability of success (p)
raised to the r power,
3. the probability of failure (q)
raised to the (n - r) power.
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The
probability of an event, p, occurring
exactly r
times: |
n = number of trials
r = number of specific events
you wish to
obtain
p = probability that the event
will occur
q = probability that the event will
not occur
(q = 1 - p, the
complement of the event) |
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Alternative formula form |
The graphing of all possible binomial
probabilities related to an event creates a binomial
distribution. Consider the following distributions of tossing a
fair coin:
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Two Tosses |
Four Tosses |
Examples: (answers rounded
to three decimal places)
1.
A test consists of 10 multiple choice
questions with five choices for each question. As an
experiment, you GUESS on each and every answer without even reading
the questions.
What is the probability of getting exactly 6 questions
correct on this test? |
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Solution:
n = 10
r = 6
n – r = 4
p = 0.20 = probability of guessing the correct
answer on a question
q = 1 - p = 0.80 = probability of not
guessing the correct answer on a question
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| 2.
When rolling a die 100
times, what is the probability of rolling a "4"
exactly 25 times? |
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Solution:
n = 100
r = 25
n – r = 75
p = 1/6 = probability of rolling a "4"
q = 1 - p = 5/6 = probability of not
rolling a "4"
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| 3.
At a certain intersection, the light for eastbound traffic
is red for 15 seconds, yellow for 5 seconds, and green for
30 seconds. Find the probability that out of the next
eight eastbound cars that arrive randomly at the light,
exactly three will be stopped
by a red light. |
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Solution:
n = 8
r = 3
n – r = 5
p = 15/50 = probability of a red light
q = 1 - p = 35/50 = probability of not a red light
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How to use your
TI-83+/84+ graphing calculator with Bernoulli Trials.
Click calculator. |
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