Normal approximations to binomial distributions.

1. Binomial Distributions
or quot;Will this SUCCEED or FAIL?
352/365: Jumpin' for Mikey
by flickr user Mr.Thomas

2. There are five children in a family. Assume that boys and girls are
equally likely.
HOMEWORK
1. What is the probability that three are girls?
binompdf(5, 1/2, 3)
2. What is the probability that there are at most three girls
(i.e., there may be 0, 1, 2, or 3 girls)?
binomcdf(5, 1/2, 3)
3. What is the probability that there are more than two girls?
1 – binomcdf(5, 1/2, 2)

3. A shipment of 200 tires from a tire manufacturing company is
known to include 40 defective tires. Five tires are selected at
random, and each tire is replaced before the next tire is selected.
(a) What is the probability of getting at most 2 defective tires?
binomcdf(5, 40/200, 2) HOMEWORK
(b) What is the probability of getting at least 1 defective tire?
1 - binomcdf(5, 40/200, 0)
(c) What is the probability of getting 2 or 3 defective tires?
binompdf(5, 40/200, 2) + binompdf(5, 40/200, 3)

4. At a certain hospital, the probability that a newborn is a boy is
0.47. What is the probability that between 45 and 60 (inclusive) of
the next 100 babies will be boys?
HOMEWORK
binomcdf(100, 0.47, 60) — binomcdf(100, 0.47, 44)

5. Working With Binomial
Distributions
Let's take apart a typical
problem about alarm
clocks and see how the
pieces fit together.
Puzzle Alarm Clock by evadedave

6. A manufacturer produces 24 yard alarms per week. Six percent of
all the alarms produced are defective. What is the probability of
getting two defective alarms in one week?
'S' and 'F' (Success and Failure) are the possible outcomes of a
trial in a binomial experiment, and 'p' and 'q' represent the
probabilities for 'S' and 'F.'
• P(S) = p • P(F) = q = 1 - p
• n = the number of trials
• x = the number of successes in n trials
• p = probability of success
• q = probability of failure
• P(x) = probability of getting exactly x successes in n trials
Note that 'Success' in this case, is the probability of selecting a
defective alarm.

7. A manufacturer produces 24 yard alarms per week. Six percent of
all the alarms produced are defective. What is the probability of
getting two defective alarms in one week?
'S' and 'F' (Success and Failure) are the possible outcomes of a
trial in a binomial experiment, and 'p' and 'q' represent the
probabilities for 'S' and 'F.'
• P(S) = p • P(F) = q = 1 - p
So how do we answer
this question?
• n = the number of trials
• x = the number of successes in n trials
• p = probability of success
• q = probability of failure
• P(x) = probability of getting exactly x successes in n trials
Note that 'Success' in this case, is the probability of selecting a
defective alarm.

8. A manufacturer produces 24 yard alarms per week. Six percent of
all the alarms produced are defective. What is the probability of
getting two defective alarms in one week?
binompdf(trials, p, x [this is optional])
trials = number of trials
p = P(success)
x = specific outcome

9. Now you try ...
Elaine is an insurance agent. The probability that she will sell a
life insurance policy to a family she visits is 0.7 (she's a really
GOOD sales lady).
(a) If she sees 8 families today, what is the probability that she
will sell exactly 5 policies?

10. Now you try ...
Elaine is an insurance agent. The probability that she will sell a
life insurance policy to a family she visits is 0.7 (she's a really
GOOD sales lady).
(b) If she sees 8 families today, what is the probability that she
will sell at most 5 policies?

11. The Binomial Coin Experiment
http://www.math.uah.edu/stat/applets/BinomialCoinExperiment.xhtml

12. Normal Approximation to
the Binomial Distribution
We have seen that binomial distributions and their histograms
are similar to normal distributions. In certain cases, a binomial
distribution is a reasonable approximation of a normal
distribution. How can we tell when this is true?

13. Normal Approximation to
the Binomial Distribution
Recall:
In a normal distribution, we used values for μ and σ to solve
problems, where:
• μ = the population mean, and
• σ = the standard deviation
In a binomial distribution, we used values for 'n' and 'p' to solve
problems, where:
• n = number of trials, and
• p = probability of success

14. Normal Approximation to
the Binomial Distribution
We now want to use the normal Link by flickr user jontintinjordan
approximation of a binomial distribution.
The distribution will be approximately normal if:
np ≥ 5 and nq≥ 5
th
be is is
Once we know that a binomial
typ twe the
es en LIN
distribution can be approximated
of the K
by a normal curve we can calculate
dis se
tri tw
the values of μ and σ like this:
bu o
tio
ns

15. Normal Approximation to
An Example
the Binomial Distribution
Border patrol officers estimate that 10 percent of the vehicles
crossing the US - Canada border carry undeclared goods. One day
the officers searched 350 randomly selected vehicles. What is the
probability that 40 or more vehicles carried undeclared goods?
Is this binomial distribution approximately normal?
What is n? Is np ≥ 5? What is μ?
What is p?
Is nq ≥ 5?
What is σ?
What is q?

16. Are the following distributions normal approximations of binomial
distributions? How do you know?
(a) 60 trials where the probability (b) 60 trials where the probability
of success on each trial is 0.05 of success on each trial is 0.20
(c) 600 trials where the probability (d) 80 trials where the probability
of success on each trial is 0.05 of success on each trial is 0.99

17. Determine the mean and standard deviation for each binomial
distribution. Assume that each distribution is a reasonable
HOMEWORK
approximation to a normal distribution.
(a) 50 trials where the probability of success for each trial is 0.35
(b) 44 trials where the probability of failure for each trial is 0.28
(c) The probability of the Espro I engine failing in less than 50 000
km is 0.08. In 1998, 16 000 engines were produced. Find the mean
and standard deviation for the engines that did not fail.

18. Solve the following problem using a binomial solution
A laboratory supply company breeds rats for lab testing. Assume that
male and female rats are equally likely to be born.
HOMEWORK
(a) What is the probability that of 240 animals born, exactly 110
will be female?
(b) What is the probability that of 240 animals born, 110 or more
will be female?
(c) What is the probability that of 240 animals born, 120 or more
will be female?
(d) Is it correct to say that, in the above situation,
P(x ≥ 120) = P(x > 119), or do we need to account
for the values between 119 and 120?

19. HOMEWORK
The probability that a student owns a CD player is 3/5. If eight
students are selected at random, what is the probability that:
(a) exactly four of them own a CD player?
(b) all of them own a CD player?
(c) none of them own a CD player?

20. HOMEWORK
The probability that a motorist will use a credit card for gas
purchases at a large service station on the Trans Canada
Highway is 7/8. If eight cars pull up to the gas pumps, what is the
probability that:
(a) seven of them will use a credit card?
(b) four of them will use a credit card?