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MAT109 Probability & Statistics
The binomial probability applies to experiments for which the
following is true:
VOCABULARY & NOTATION
binomial: comprised of only two categories
ex. binomial number system has only 1's and 0's
n
# of trials
x
# of successes
f p= f
Binomial Probability
a) the experiment consists of repeated trials
b) each trial is independent
c) each trial has only two possible outcomes
p
p = probability of success in one trial;
p x = probability of x successes occurring
Situations in the real world which have binomial probabilities
are flipping a coin (head or tail), hitting a bulls-eye (hit or
miss), making a foul shot (yes or no), and birth of a child
(b
l)
q = probability of failure in one trial;
n x
The binomial probability function is a handy formula to calculate probabilities in binomial situations.
n
P (x) = f p p x qn - x
x
Let's break this down:
P (number of successes) = f
Example 1
n = # trials
x = # successes
p f
prob of
success
# of success
p
prob of
f
p
failure
# of failures
A balanced coin is tossed 4 times. What is the probability of getting 3 heads?
n = 4 tosses
x = successes, i.e., heads = 3
p = 2 , probability of getting a head in one toss
4 13
1 4- 3
P (3heads in 4 tosses) = f p f
d1 - n
p
2
3 2
4 13 1 1
= f pf
pd 2 n
3 2
= _4 i _0.125i _0.5i = 0.25or 25%
Example 2
5 cards are drawn from a deck, one at a time and then replaced.
What is the probability that at least 3 of them will be red?
First, let's determine if this is a binomial situation, using a) through c) from above.
a) repeated trials?
b) independent trials?
Yes. 1 card at a time, 5 times
Yes. Replacing the cards after they are drawn keeps each trial unaffected by
previous outcomes.
3
2
5
c) only two outcomes?
Yes. Red or not-redp _(black).
3red in5 draws i = f p d 1 n d 1 n = 0.3125
2
3 2
The total
5 1 4 1 1
It is binomial, so we can use the formula.
probability
p _4red in5 draws i = f pd n d n = 0.15625
2
4 2
is the sum
5 1 5 1 0
** The probability of each situation which is a success must be
of these, or
p _5red in 5drawsi = f pd n d n = 0.03125
calculated separately then added together.
MAT109 Probability & Statistics
Math Center Miller Library, x7862
5
2
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Binomial Probability
http://mathcenter.washcoll.edu