Binomial Random Variables

Example 3 | Example 4| Example 5 |Useful Web Resources| Solutions

 

 

Bernoulli Trial:

An experiment that has only two possible outcomes.Eg a single toss of a coin.

  • We are usually interested in studying experiments that consist of a sequence of independent and identical Bernolli trials
  • A value of 1 is assigned to one of the two outcomes, and 0 to the other outcome. For example, 1 is assigned to flipping a Head and 0 to flipping a Tails (Such a random variable is called a Bernoulli random variable)
  • A value of 1 is commonly assigned to a success and the value of 0 is commonly assigned to a failure.

     

Example 1:

The following are sequences of independent and identical Bernoulli trials:

Let X be the number of successes.

i) Roll a dice 20 times. Each roll is independent of one another, and the probability of flipping a 4 is 1/6 on each trial.

ii)Flip a coing 40 times. Each flip of the coin is independent of one another, and the probability of flipping a Head is 1/2 on each trial.

iii)Test 100 randomly selected electronic components from manufacturing machine. If glitches in the components occur independently of one another and the probability of a glitch stays constant, then this experiment is a sequence of independent and identically distributed Bernoulli trials.

Definition 3.1.1:

A binomial random variable is the number of successes in n independent and identical Bernoulli trials.

The probability mass function of a binomial random variable is called a binomial distribution.

 

Example 2:

Experiment
Random Variable
Success
Failure
n
p
Toss a coin 10 times
X=number of heads
Head
Tails
10
1/2
Flip a dice 6 times
X=number of even numbers
2,4,6
1,3,6
6
1/6
A mutiple choice test consisting of 10 questions (Each with four answers to choose from)
X=number of correct answers
Correct answer
Incorrect answer
10
1/4

Theorem 3.1-1:

Let Y be a binomial random variable based on n independent and identical Bernoulli trails with probability of a success p. Then,

 

Example 3:

A mutiple choice consists of 15 questions, each of which have 4 choices to choose from. The Y denote the total number of correct answers. What is the probability of getting a 60% on the test?

Solution:

60% of 15 is 9. So you must get 9 questions correct, to get a 60% on the test. The probability of getting a correct answer is 1/4=0.25 because only one of the four choices are correct. Therefore the probability of an incorrect answer is 3/4= 0.75.

P(Y=9) = (15C9) (0.25)^9 (0.75)^ 6=0.003398=0.0034

    What is P(Y=12)?

    1. 0.225
    2. 0.00001144
    3. 1
    4. 0.0032

    What is P(Y>13)?

    1. 8.8 x 10^(-7)
    2. 0.1559
    3. 5.6810 x 10^(-8)
    4. 0.00675

Solution

 

Theorem 3.1-2:

The mean and variance of a binomial random variable Y with parameters n and p are

E(Y)= np and

VAR(Y) = np(1-p)

 

This theroem can be verified by expressing Y as a sum of n independent Bernoulli random variables. That is Y=SXi, where the Xi's are independent with P(Xi =1) = p and P(Xi = 0) = 1-p

     

Example 4:

Let Y denote the number of 5's in 72 tosses of a dice. Then E(Y)= 72(1/6)=12. That is, we would observe 12 5's in 72 tosses of a dice. VAR(Y)=72(1/6)(5/6)=10, and STD(Y)= 3.16.

Assume that you are rolling an unfair dice. The probability of rolling each number is given in the table below. Let Y denote the number of 2's in 50 tosses of a dice.

 

y
1
2
3
4
5
6
p(y) 0.1 0.2 0.1 0.2 0.1 0.3

 

What is E(Y)?

What is VAR(Y)?

Solution

 

The shape of the binomial distribution depends on the value of the parameters n and p. If p=1/2, then the distribution is symetric about n/2. For p!=1/2 (p not equal to 1/2), the distribution is asymetric. The degree of asymetry increases as p approaches 0 or 1 and decreases as n increases.Use the applet to verify these characteristics. Help!

 

Example 5:

Let Y be a binomial random variable with n= 8 and p= 0.3. Using table 2 in Appendix A of the text book, we find that

P(Y>5)=1-P(Y£5) =1- 0.9887=0.0113

P( 2 £ Y £ 5)= P(Y£5) - P(Y£2)=0.9887-0.5518=0.4369

 

Round all answers to 4 decimal places.

n
6
9
20
p
0.65
0.45
0.15
Range
P(Y>2)
P(3 £ Y£ 7)
P(Y£6)
Answer

 

Solution

 

This applet can also be used to calculate cumulative probabilities of binomial random variables for given values of n and p. Use the applet to calculate the following probabilities. Help!

Round all answers to 4 decimal places.

n
100
60
300
p
0.33
0.21
0.60
Range
P(Y=<49)
P( 11£ Y £ 47)
P(Y>=180)
Answer

Solution

Useful Web Resources

Bernoulli Trials and Binomial Random Variables

Random Variables and Distributions

The Bernoulli Random Variable -www.math.mcmaster.ca/canty/teaching/stat2d03/lectures4.pdf

 

Solutions

 

Solution to Example 3

A: Since order does not matter the we use 15C12 to find the total number of ways we can have 12 correct answers and 3 incorrect answers. The probability that 12 questions are correct is (0.25)^12, and the probability that the 3 remaining questions are incorrect is (0.75) ^3.

So we get P(Y=12)= (15C12)(0.25)^12 (0.75)^3 =0.00001144=1.144 x 10^(-5)

B: P(Y>13)= P(Y=14) + P(Y=15)= (15C14)(0.25)^14 (0.75) + (15C15)(0.25)^15=5.6810 x 10^(-8)

Solution to Example 4

A: P(Y=2)= p =0.2

So 1-p = 1-0.2 = 0.8

So E(Y)=50(0.2)=10 and

B: VAR(Y)=50(0.2)(0.8)= 8

Solution to Example 5A

P(Y>2)=1-P(Y£2)= 1-0.11740= 0.8826

P(3 £ Y£ 7) =P(Y£ 7) - P(Y£ 3) =0.9909-0.3614=0.6295

P(Y>6)=1-P(Y£ 6)=1-0.9781 =0.0219

 

Solution to Example 5B

P(Y=<49)=0.9996

P( 11£ Y £ 47) =0.7413

P(Y>=180)=0.5250

HELP

Click on the icon at the side and a Web page will open up. Scroll down until you find an applet labeled Binomial Distribution. Enter a value for n and p and press Rescale. The diagram will change accordingly.

HELP

For n=100, p=0.33 find P(Y=<49):

Enter n=100 and p=0.33 into the boxes labeled n and p respectively on the applet. Click Rescale. Click on the button labeled Prob found at the top of the applet, and select x<=b. A box will appear asking for the user to input an upper bound. Enter 49 and click OK. The diagram highlight the range specified in red. The probability of that range will appear in red at the bottom of the applet.

The other questions are done similarly.

 

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