Probability Explanation And Exercises Worksheet Page 21
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37"
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!
!
probability of 0.5 of being a success on each trial. This makes Figure 1 an example
P(6!or!head)!=!P(6)!+!P(head)!.!P(6!and!head)!
head)!=!P(6)!+!P(head)!.!P(6!and!head)!
of a binomial distribution.
!!!!!!!!!!!!!=!(1/6)!+!(1/2)!.!(1/6)(1/2)!
!=!(1/6)!+!(1/2)!.!(1/6)(1/2)!
The Formula for Binomial Probabilities
!!!!!!!!!!!!!=!7/12!
!=!7/12!
The binomial distribution consists of the probabilities of each of the possible
!
numbers of successes on N trials for independent events that each have a
probability of π (the Greek letter pi) of occurring. For the coin flip example, N = 2
Binomial&Distributions&
ial&Distributions&
and π = 0.5. The formula for the binomial distribution is shown below:
!
!
( ) =
( 1
)
!
( ) =
( 1
)
!
! (
) !
! (
) !
&
where P(x) is the probability of x successes out of N trials, N is the number of
trials, and π is the probability of success on a given trial. Applying this to the coin
&
flip example,
2!
2
2!
2
( 0 ) =
( . 5 ) (1
( 1 )( . 25 ) = 0.25!
.5)
=
( 0 ) =
( . 5 ) (1
( 1 )( . 25 ) = 0.25!
.5)
=
0! ( 2
0 ) !
2
0! ( 2
0 ) !
2
2!
2
2!
2
( 1 ) =
( . 5 ) (1
( . 5 )( . 5 ) = 0.50!
.5)
=
( 1 ) =
( . 5 ) (1
( . 5 )( . 5 ) = 0.50!
.5)
=
1! ( 2
1 ) !
1
1! ( 2
1 ) !
1
2!
2
2!
2
( 2 ) =
( . 5 ) (1
( . 25 )( 1 ) = 0.25!
.5)
=
( 2 ) =
( . 5 ) (1
( . 25 )( 1 ) = 0.25!
.5)
=
2! ( 2
2 ) !
2
2! ( 2
2 ) !
2
If you flip a coin twice, what is the probability of getting one or more heads? Since
!
the probability of getting exactly one head is 0.50 and the probability of getting
exactly two heads is 0.25, the probability of getting one or more heads is 0.50 +
0.25 = 0.75.
Now suppose that the coin is biased. The probability of heads is only 0.4.
What is the probability of getting heads at least once in two tosses? Substituting
into the general formula above, you should obtain the answer .64.
Cumulative Probabilities
We toss a coin 12 times. What is the probability that we get from 0 to 3 heads? The
answer is found by computing the probability of exactly 0 heads, exactly 1 head,
exactly 2 heads, and exactly 3 heads. The probability of getting from 0 to 3 heads
205
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