Introduction to Binomial Distributions.

1. Binomial
Distributions
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2. An orange producer who calls himself Doctor Juice grows an exclusive
variety of oranges which are sorted into three categories and sold at
different prices.
Description Size Price per orange
Small less than 75mm 12 cents
Jumbo largest 12% 45 cents
Regular all others 35 cents
The diameters of the oranges are distributed normally with a mean
HOMEWORK
of 84 mm and a standard deviation of 12 mm.
(a) What percent of the oranges are sorted into the small
category?
Hand this in now
(b) What is the minimum diameter (rounded to the nearest
millmeter) of a Jumbo Orange?
(c) What is the expected income from 2000 unsorted oranges,

3. The life of a toaster is found to be normally distributed with a mean life of
4.7 years and a standard deviation of 0.9 years. The manufacturer of the
toaster will replace free of charge any toaster that malfunctions while
under warranty. For how many months should the manufacturer
guarantee the toasters if no more than 10 percent of them are to be
replaced? Will the manufacturer likely replace 10 percent of the toasters?

4. Binomial
Distributions
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5. In some probability experiments, there are exactly two possible
outcomes. For example:
• When flipping a coin, you get heads or tails. When flipping a
coin 200 times, what is the probability of getting heads exactly
101 times? (The answer is 5.58 percent)

6. In some probability experiments, there are exactly two possible
outcomes. For example:
• When flipping a coin, you get heads or tails. When flipping a
coin 200 times, what is the probability of getting heads exactly
101 times? (The answer is 5.58 percent)
• When answering a question on a multiple choice test, your
answer will be right or wrong. What is the probability of getting 50
percent of the answers correct if there are four choices for each
question, and you guess each answer?

7. In some probability experiments, there are exactly two possible
outcomes. For example:
• When flipping a coin, you get heads or tails. When flipping a
coin 200 times, what is the probability of getting heads exactly
101 times? (The answer is 5.58 percent)
• When answering a question on a multiple choice test, your
answer will be right or wrong. What is the probability of getting 50
percent of the answers correct if there are four choices for each
question, and you guess each answer?
• When a manufacturer guarantees a toaster for one year, the
toaster will work fine or fail in one year. The manufacturer knows
that, on average, 5 percent of the toasters fail. What is the
probability that the manufacturer will have to replace 120 or more
toasters in a year when 2000 toasters are sold?

8. In some probability experiments, there are exactly two possible
outcomes. For example:
• When flipping a coin, you get heads or tails. When flipping a
coin 200 times, what is the probability of getting heads exactly
101 times? (The answer is 5.58 percent)
Binomial
• When answering a question on a multiple choice test, your
answer will be right or wrong. What is the probability of getting 50
Distributions
percent of the answers correct if there are four choices for each
question, and you guess each answer?
• When a manufacturer guarantees a toaster for one year, the
toaster will work fine or fail in one year. The manufacturer knows
that, on average, 5 percent of the toasters fail. What is the
probability that the manufacturer will have to replace 120 or more
toasters in a year when 2000 toasters are sold?

9. Type of Distributions...
Uniform Distribution: data may be discrete or continous. Every
outcome in the experiment is equally likely.
Example: graph the distribution that shows what can happen
when a 6-sided die is thrown.
No data between
Probabilities of outcomes 0 and 1.
when rolling a six sided
die.
Uniform (Probability)
Distribution

10. Type of Distributions...
Normal Distributions: Data is continous (height, weight, time, etc.)
when certain experiments are carried out many, many, many times
the probability graph of the data tend to be quot;bell shapedquot; this is
known as the Normal Curve.

11. How many girls are there
in a family of four children.
Find the probability of there being
0, 1, 2, 3, or 4 girls in the family.

12. Type of Distributions...
Binomial Distribution: data is discrete (# of heads when ten
coins are tossed, # of spades in a 13 card hand , etc.). When a
binomial experiment is conducted many, many, many times a
portion of the related histogram approaches the shape of the
normal curve.
Probability of the
number of girls in a
family of four.
Experimental Binomial
(Probability)
Theoretical Binomial
Distribution
(Probability)
Distribution

13. Theoretical Binomial
(Probability) Distribution
binompdf(trials, p, x [this is optional])
trials = number of trials
p = P(success)
x = specific outcome

14. The weights of babies born in a certain hospital average 8 lb 1 oz, with
a standard deviation of 12 oz. Assume that the weights are normally
distributed.
HOMEWORK
(a) Find the percentage of babies with a birth weight between 7
and 9 pounds.
(b) Find the weight, W, such that the percentage of babies with a
birth weight greater than W is 60 percent.
(c) Find the weight, W, such that the percentage of babies with a
birth weight less than W is 25 percent.

15. A college aptitude test is scaled so that its scores approximate a normal
distribution with a mean of 500 and standard deviation of 100.
(a) Find the probability that a student selected at random will
HOMEWORK
score 800 or more points.
(b) Find the score x, such that 76 percent of the students have a
score less than x.

16. Forty students measured the width of the gym, and wrote their
measurements in centimetres, rounded to the nearest cm. The
measurements are recorded on the table below.
HOMEWORK
2251 2249 2250 2247 2253 2248 2249 2253
2254 2247 2250 2253 2248 2255 2249 2249
2250 2251 2252 2250 2249 2250 2247 2250
2250 2252 2253 2255 2254 2248 2248 2242
2249 2245 2251 2246 2250 2246 2251 2246
Draw a histogram of the data. Using the properties of a Normal
Distribution, determine if the data is approximately normal.