Download 5.4 Binomial Random Variable

2/5/2016
5.4 ­ Binomial Random Variable | STAT 200
STAT 200
Elementary Statistics
5.4 ­ Binomial Random Variable
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Binomial random variable: A specific type of discrete random variable that counts how often a
particular event occurs in a fixed number of tries or trials
For a variable to be a binomial random variable, ALL of the following conditions must be met:
1. There are a fixed number of trials (a fixed sample size)
2. On each trial, the event of interest either occurs or does not
3. The probability of occurrence (or not) is the same on each trial
4. Trials are independent of one another
Examples of Binomial Random Variables
Number of correct guesses at 30 true­false questions when you randomly guess all
answers
Number of winning lottery tickets when you buy 10 tickets of the same kind
Number of left­handers in a randomly selected sample of 100 unrelated people
Number of tails when flipping a coin 10 times
Notation
n = number of trials
p = probability event of interest occurs on any one trial
Example
Number of correct guesses at 30 true­false questions when you randomly guess all answers
There are 30 trials, therefore n = 30
There are two possible outcomes (true and false) that are equally probable, therefore p = 1/2
= .5
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Probabilities for Binomial Random Variables
The conditions for being a binomial variable lead to a somewhat complicated formula for finding the
probability any specific value occurs (such as the probability you get 20 right when you guess as 30
True­False questions.)
We'll use Minitab Express to find probabilities for binomial random variables. However, for those of
you who are curious, the by hand formula for the probability of getting a specific outcome in a
binomial experiment is:
Binomial Random Variable Probability
[Math Processing Error]
n = number of trials
x = number of successes
p = probability event of interest occurs on any one trial
! is the symbol for factorial. For a review of factorials, see the course algebra review page
(https://onlinecourses.science.psu.edu/statprogram/node/199) .
One can use the formula to find the probability or alternatively, use Minitab Express to find the
probability. In the homework, you may use the method that you are more comfortable with unless
specified otherwise.
In the following Minitab Express example we will find P(x) for n = 20, x =3, and p = 0.4
Using Minitab Express (#) Using Minitab (#)
To calculate binomial random variable probabilities in Minitab:
1. Open Minitab without data.
2. From the menu bar select Calc > Probability Distributions > Binomial.
3. Choose Probability since we want to find the probability x = 3.
4. Enter 20 in the text box for number of trials.
5. Enter 0.4 in the text box for probability of success (note for Minitab versions over 14
this now labeled event probability).
6. Since we do not have a column of data select the radio button for Input Constant and
enter 3.
7. Click Ok.
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Minitab output:
Probability Density Function
Binomial with n = 20 and p = 0.4
x
P(X = x)
3.00
0.0123
Video Review
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To calculate binomial random variable probabilities in Minitab Express:
1. Open Minitab Express without data.
2. From the menu bar, select Statistics > Probability Distributions > CDF/PDF >
Probability (PDF).
3. Since we want to find the probability that x = 3, enter 3 into the "Value" box
4. In the "Distribution" drop down menu, select Binomial.
5. Enter 20 into the "Number of trials" box, and 0.4 into the "Event probability" box.
6. Select "Display a table of probability density values" to show the output.
7. Click Ok
The result should be the following output:
Video Review
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In the following example, we illustrate how to use the formula to compute binomial probabilities by
hand. If you don't like to use the formula, you can also use Minitab Express to find the probabilities.
Example
Red Flowers
Cross­fertilizing a red and a white flower produces red flowers
25% of the time. Now we cross­fertilize five pairs of red and
white flowers and produce five offspring. Find the probability
that there will be no red flowered plants in the five offspring.
X = # of red flowered plants in the five offspring.
The number of red flowered plants has a binomial distribution with n = 5, p = .25
[Math Processing Error]
There is a 23.7% chance that none of the five plants will be red flowered.
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Cumulative probability: Likelihood that a certain number of successes or fewer will occur.
Binomial random variable probabilities are mutually exclusive, therefore we can use the addition rule
that we learned in Lesson 4.
Example
Red Flowers, cont.
Continuing with the red flowers example, what if we wanted to know the probability that
there would be one or fewer red flowered plants?
[Math Processing Error]
There is a 63.2% chance that one or fewer of the five plants will be red flowered.
In the red flowers example, we first computed P(X = x) and then P(X ≤ x). This latter expression is
called finding a cumulative probability because you are finding the probability that has accumulated
from the minimum to some point, i.e. from 0 to 1 in this example
To use Minitab Express to solve a cumulative probability binomial problem, return to Statistics >
Probability Distributions> CDF/PDF > Cumulative Distribution Function (CDF). For Value enter 1.
For distribution select the binomial. There are 5 trials and the event probability is .25
To use Minitab to solve a cumulative probability binomial problem, return to Calc > Probability
Distributions > Binomial as shown above. Now however, select the radio button for Cumulative
Probability. For Number of Trials enter 5 and the event probability is .25. Click the radio button for
Input Constant and enter the x value of 1.
Expected Value and Standard Deviation for Binomial Random
Variable
The formula given earlier for discrete random variables could be used, but the good news is that for
binomial random variables a shortcut formula for expected value (the mean) and standard deviation
can also be used.
Bionomial Random Variable Formulas
[Math Processing Error]
[Math Processing Error]
n = number of trials
p = probability event of interest occurs on any one trial
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After you use this formula a couple of times, you'll realize this formula matches your intuition. For
instance, the “expected” number of correct (random) guesses at 30 True­False questions is np = (30)
(.5) = 15 (half of the questions). For a fair six­sided die rolled 60 times, the expected value of the
number of times a “1” is tossed is np = (60)(1/6) = 10.
The standard deviations for these would be, for the True­False test, [Math Processing Error], and for
the die, [Math Processing Error].
Example
Roulette
A roulette wheel has 38 slots, 18 are red, 18 are black, and 2
are green.You play five games and always bet on red.
How many games can you expect to win?
Recall, you play five games and always bet on red. [Math
Processing Error] and [Math Processing Error]
[Math Processing Error]
[Math Processing Error]
Out of 5 games, you can expect to win 2.3684 (with a standard deviation of 1.1165).
What is the probability that you will win all five games?
[Math Processing Error]
[Math Processing Error]
[Math Processing Error]
There is a 2.38% chance that you will win all five out of five games.
If you win three or more games, you make a profit. If you win two or fewer games, you
lose money. What is the probability that you will win no more than two games?
[Math Processing Error]
[Math Processing Error]
[Math Processing Error]
[Math Processing Error]
[Math Processing Error]
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There is a 54.93% chance that you will win no more than two games. In other words, there
is a 54.93% chance that you will lose money.
Video Review: Working with Binomial Random Variables
Binomial Random Variable: Flipping a Coin Example
A fair coin is flipped 10 times. This example is used to demonstrate calculations concerning
binomial random variables. Hand calculations are performed and Minitab Express is used.
Binomial Random Variable: Guessing on a Multiple Choice Quiz Example
A class is taking a multiple choice quiz. There are 6 questions, each with 4 options. The
professor accidently brought a quiz from a different, much more advanced class. All
students randomly guess on each item. This is a binomial random variable. We compute the
mean and standard deviation by hand. We compute the probability that a student will pass
(i.e., at least 60%), the probability that they will get all questions incorrect, and the
probability that they will get all questions correct all using Minitab Express.
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‹ 5.3 ­ Expected Value of a Discrete
Random Variable (/stat200/node/36)
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