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Construct a binomial probability distribution.
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Find binomial probabilities with the binomial formula.
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Find the graph, mean, variance, and standard deviation.
Notes
Binomial Experiments
A binomail experiment is a probability experiment that satisfies the
conditions:
1. The experiment is repeated for a number of independent trials, n.
2. Each trial has two possible outcomes: success (S) or failure (F).
3. The probability for success p is the same for each trial.
q = 1 − p is the probability for failure for each trial.
4. The binomial random variable x counts the number of successful
trials.
If a probability experiment is binomial, you will be able to specify
values for n, p, and q. Which of the following are binomial
experiments? Give n, p, and q and list the possible values for x.
Otherwise, explain why it is not a binomail experiment.
Notes
1. A surgical procedure has an 85% chance of success. A doctor
performs the procedure on eight patients. The random variable
represents the number of successes.
2. A jar contains 5 red marbles, 9 blue marbles, and 6 green marbles.
You randomly select three marbles from the jar without replacement.
The random variable represents the number of red marbles.
3. U.S. adults using Twitter were asked if they ever posted comments
about their personal lives. The results were yes (48%) and no (52%).
You randomly select seven U.S. adults using Twitter. The random
variable is the number of people that respond ”yes.”
We have learned to count the number of combinations of n objects
taken r at a time is
n!
n Cr =
(n − r )!r !
With binomial experiements, we are using x to count the number of
successes, regardless of order, taken from n trials, and this is exactly
n Cx
=
n!
(n − x)!x!
The probability of a single simple event with x successes and n − x
failures, is
p x q n−x
There are outcomes where the number of successes x are the same,
but the order is different. The total probability is then
P(x) = n Cx p x q n−x =
n!
p x q n−x
(n − x)!x!
Notes
Example
Notes
A survey indicates that 41% of women in the United States consider
reading their favorite leisure-time activity. You randomly select four
U.S. women and ask them if reading is their favorite leisure-time
activity. The probability distribution is built through the formula:
Using n = 4, p = 0.41, q = 0.59 for x = 0, meaning none of the
women responded with yes, the probability is
P(x = 0) = 4 C0 (0.41)0 (0.59)4 = 1(0.41)0 (0.59)4 ≈ 0.121
Continuing for x = 1, x = 2, x = 3, x = 4,
P(x = 1) = 4 C1 (0.41)1 (0.59)3 = 4(0.41)1 (0.59)3 ≈ 0.337
P(x = 2) = 4 C2 (0.41)2 (0.59)2 = 6(0.41)2 (0.59)2 ≈ 0.351
P(x = 3) = 4 C3 (0.41)3 (0.59)1 = 4(0.41)3 (0.59)1 ≈ 0.163
P(x = 4) = 4 C4 (0.41)4 (0.59)0 = 1(0.41)4 (0.59)0 ≈ 0.028
Example
Notes
A survey indicates that 21% of men in the United States consider
fishing their favorite leisure-time activity. You randomly select five
U.S. women and ask them if fishing is their favorite leisure-time
activity. Build the probability distribution.
Although the formulas for mean, variance, and standard devaition can
be used for binomial distributions, knowing the parameters n, p, q
make the formulas much simpler.
Population parameters of a binomial distribution
Mean: µ = np
Variance: σ 2 = npq
√
Standard deviation: σ = npq
Ex: A surgical procedure has an 85% chance of success. A doctor
performs the procdure on eight patients. On average, there are
µ = 6.8 successful surgeries performed. The standard deviation is
σ ≈ 1.01. It would be unusual if values are more than two standard
deviations away. That means fewer than 6.8 − 2(1.01) = 4.78
successes is unusual.
Notes
Assignment
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Homework §4.2: 1, 2, 18, 20, 26, 31, 33
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Study Guide: Chapter Quiz 4: 1, 2, 3
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Extra review: Page 229: 19, 23, 25, 27
Notes
Notes
Notes