Download Binomial Probability and Distribution

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Unit 7: Probability Distributions
Ms. Sanchez
+ Binomial
Experiments
Is
a probability experiment that satisfies
these conditions.
 There
exist a fixed number of trials, where
each trial is independent of the others.
Only
two outcomes: one is always given
p
= success p=1 – q
 q = failure q= 1 - p
Central
problem of a binomial experiment
is to find the probability of x successes out
of n trials.
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Notation
n: total
number of trials
p: probability
p
=1– q
q: probability
q
of success in a single trial
of failure in a single trial
=1– p
X: the
random variables represent a count
of the number of successes in n trials. [ie:
possible success in the total trials]
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Binomial Theorem
The probability of
exactly x successes
in n trials is
P(x) = (nCx)(p )(q
x
n-x
)
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Example
5
contestants have a trail at the wheel of
fortune. Say landing on gold is success
and not landing on gold is failure. There
is a 2% chance of landing on gold.
 What
is the probability of exactly 3
successes?
 What is the probability of 3 or more
successes?
 What is the probability that less than half of
the contestants succeed?
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E #1
Biologist are studying a new
hybrid tomato. The probability of
germinating is 0.70. The biologist
plant 6 seeds and want to know
the probability of 4 of them
germinating
What
if they plant 8 seeds?
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Graphing a Binomial
Distribution
A
waiter knows that 7 out of 10 customers will leave
a good tip when dining alone. During lunch hour he
serves 5 people total. Find the Binomial Distribution
of her lunch shift.
1.
Probabilities of all possible outcomes, that the
patrons will leave a good tip [0, 1, 2, 3, 4, 5]
2.
Then construct a histogram.
a.
Place X values on the horizontal axis.
b.
Place P(x) values on the vertical axis.
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E # 3. Find the probabilities of all
outcomes, and create the histogram.
1.
When Melecio practices penalty kicks, he
makes 80% of them. For practice he kicks 8,
draw the binomial distribution.
2.
A satellite is powered by 5 solar cells. The
probability that one cell won’t fail is 0.75.
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Review of Normal Distribution
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Binomial approximation to Normal
Distribution.
Mean: is
the expected number of
successes
Standard
m = np
Deviation:
s = npq
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Example: Binomial to Normal
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When Cesar practices penalty kicks, he makes 80% of them.
For practice he kicks 8, draw the binomial distribution.
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Exercise #1: Draw the Binomial
approximation to Normal
1.
N = 100 p = 0.70
2.
A satellite is powered by 10 solar cells. The
probability that one cell won’t fail is 0.65.
3.
Biologist are studying a new hybrid tomato.
The probability of germinating is 0.45. The
biologist plant 100 seeds
+ Geometric
Distribution
Geometric Distribution is a discrete
probability distribution that satisfies these
conditions.
 A trial is repeated until a success occurs.
 The trials are independent of each other
 P is the probability of success.
 X represents the number of trials in
which the first success occurs.
The probability that the 1st success will
occur on trial x is
x-1
P(x) = pq
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Example: Geometric
Distribution
A
patient is waiting for a suitable
matching kidney donor. If the
probability that a randomly selected
donor is a suitable match is p=0.1. What
is the probability that the patient will get
a suitable donor after 3 attempts?
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