Transcript SRWColAlg6_09_03

College Algebra
Sixth Edition
James Stewart  Lothar Redlin

Saleem Watson
9
Probability
and Statistics
9.3
Binomial Probability
Probability Involving a Weighted Coin
A coin is weighted so that the probability of
heads is 0.6.
• What is the probability of getting exactly
two heads in five tosses of this coin?
Probability Involving a Weighted Coin
Since the tosses are independent, the
probability of getting two heads followed
by three tails is
Probability Involving a Weighted Coin
But this is not the only way we can get exactly
two heads.
• The two heads could occur, for example,
on the second toss and the last toss.
Probability Involving a Weighted Coin
In this case, the probability is
Probability Involving a Weighted Coin
In fact, the two heads could occur on any two
of the five tosses.
• Thus, there are C(5, 2) ways in which this can
happen, each with probability (0.6)2(0.4)3.
• It follows that
P (exactly 2 heads in 5 tosses)  C(5,2) 0.6  0.4 
2
 0.023
3
Binomial Experiments
The probabilities that we have just calculated
are examples of binomial probabilities.
• In general, a binomial experiment is one in which
there are two outcomes, which we call “success” and
“failure”.
• In the coin-tossing experiment described previously,
“success” is getting “heads” and “failure” is getting
“tails”.
• The following tells us how to calculate the probabilities
associated with binomial experiments when we perform
them many times.
Binomial Probability
An experiment has two possible outcomes.
• S and F – called “success” and “failure”
• With P(S) = p and P(F) = 1 – p.
• The probability of getting exactly r successes
in n independent trials of the experiment is
P(r successes in n trials) = C(n, r)pr(1 – p)n – r
Binomial Probability and Binomial Coefficient
The name “binomial probability” is appropriate
because C(n, r) is the same as the binomial
n

coefficient   .
r 
E.g. 1—Binomial Probability
A fair die is rolled 10 times.
Find the probability of each event.
a) Exactly 2 sixes.
b) At most 1 six.
c) At least 2 sixes.
E.g. 1—Binomial Probability
We interpret “success” as getting a six and
“failure” as not getting a six.
• So, P(S) = 1/6 and P(F) = 5/6.
• Since each roll of the die is independent from the
others, we can use the formula for binomial
probability with n = 10, p = 1/6.
E.g. 1—Binomial Probability
Example (a)
Using these values in the formula gives us:
P(exactly 2 are sixes) = C(10, 2)(1/6)2(5/6)8
≈ 0.29
E.g. 1—Binomial Probability
Example (b)
The statement “at most 1 six” means 0 sixes
or 1 six. So
• So P(at most one six)
= P(0 sixes or 1 six)
= P(0 sixes) + P(1 six)
= C(10, 0)(1/6)0(5/6)10 + C(10, 1)(1/6)1(5/6)9
≈ 0.1615 + 0.3230
≈ 0.4845
E.g. 1—Binomial Probability
Example (c)
The statement “at least two sixes” means two
or more sixes.
• Adding the probabilities that 2, 3, 4, 5, 6, 7, 8, 9,
or 10 are sixes is a lot of work.
• It’s easier to find the probability of the complement
of this event.
– The complement of “two or more are sixes” is
“0 or 1 are sixes”.
E.g. 1—Binomial Probability
Example (c)
So
P(two or more sixes)
= 1 – P(0 or 1 six)
= 1 – 0.4845
= 0.5155
The Binomial Distribution
The Binomial Distribution
We can describe how the probabilities of an
experiment are “distributed” among all the
outcomes of an experiment by making a table
of values.
• The function that assigns to each outcome its
corresponding probability is called a probability
distribution.
• A bar graph of a probability distribution in which the
width of each bar is 1 is called a probability
histogram. The next example illustrates these
concepts.
E.g. 4—A Binomial Distribution
A fair coin is tossed eight times, and the
number of heads is observed.
• Make a table of the probability distribution, and
draw a histogram.
• What is the number of heads that is most likely to
show up?
E.g. 4—A Binomial Distribution
This is a binomial experiment with n = 8 and
p= ½, so 1 – p = ½ as well.
• We need to calculate the probability of getting 0
heads, 1 head, 2 heads, 3 heads, and so on.
E.g. 4—A Binomial Distribution
To calculate the probability of 3 heads, we
have
3
5
28
 1  1
P(3 heads )  C(8,3)     
256
2 2
E.g. 4—A Binomial Distribution
The other entries in the following table are
calculated similarly.
E.g. 4—A Binomial Distribution
We draw the histogram by making a bar for
each outcome with width 1 and height equal
to the corresponding probability.
The Binomial Distribution
Notice that the sum of the probabilities in a
probability distribution is 1.
• because the sum is the probability of the
occurrence of any outcome in the sample space
(this is the certain event).