The Binomial Distribution

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The Binomial Distribution
Permutations
How many different pairs of two items are
possible from these four letters L, M. N, P.
L,M L,N L,P M,L M,N M,P N,L N,M N,P P,L P,M P,N
P of Permutations N of items r how many
taken at a time
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Combinations
(order is not important)
L,M L,N L,P M,N M,P N,P
C of Combinations N of items r how many
taken at a time
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If we flip a single coin, there are two possible
outcomes head or tail (H or T).
P(one head) .5 P(no head) .5
Together the probabilities of these possible
outcomes sum to 1.0.
If we flip two coins there are four possible
outcomes.
H,H H,T T,H T,T
P(two heads) .25 P(one head) .50 P(no heads)
.25 Total 1.0
Can be calculated by P P(H) and Q P(T)
Probability of 2 heads
Probability of 1head and 1 tail
Or
Probability of 2 tails
Think about the frequency distribution.
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Flip 3 Coins 8 possible outcomes
P(3 heads) 1/8 or .13 P(2 heads) 3/8 or
.37 P(1 head) 3/8 or .37 P(0 heads) 1/8 or
.37 Total 1.0
H,H,H H,H,T H,T,H T,H,H T,T,H T,H,T H,T,T T,T,T
Can be calculated by..
or
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Binomial Distribution
A binomial distribution is produced when each of
a number of Independent trials results in one of
two Mutually Exclusive outcomes.
A single flip of a coin is a Bernoulli
trial. They reflect a discrete variable.
P(X) Probability of X number of an outcome N
Number of trials P Probability of success on
any given trial Q (1 P)
The number of combinations of N things taken X
at a time
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Example
Guess if I am thinking of the colour black or
white. Because I randomly choose one of the two
colours, the probability that I am thinking of
black is .5 on any given trial. There will be 10
trials. What is the probability of guessing 8 of
them correctly?
45(.009765) .0439425
This is the probability of OR MORE correct,
but of exactly 8 out of 10 correct.
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Binomial Distribution Testing a Hypothesis
Let us repeat the previous example, but now the
question is what is the probability of guessing
8 or more of the trials correctly.
We know that P(8) .44 That is less than .05,
But that is P(8) against 10, including P(9) and
P(10). We need to test the probability of 8 or
more correct.
P(8) .044 P(9) .010 P(10) .001 Total P
.055
The probability of guessing 8 or more correctly
out of 10 turns out to have a probability
greater than .05, thus we fail to reject the null
hypothesis. Which was what?
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Binomial Distribution
We can calculate the probability of 0 to 10
correct.
Correct Probability

• 0 .001
• .010
• .044
• .117
• .205
• .246
• .205
• .117
• .044
• .010
• .001

Mathematical Sampling Distribution
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As N gets large, all binomial distributions
approach normality, regardless of P.
Thus,
Mean NP
For example, 10(.5) 5 That is, if you examine
the data on the previous slide, you will see that
the mean of the distribution is 5 (the number of
correct trials).
Variance NPQ
For example, 10(.5)(.05) 2.25
Standard Deviation
1.58