Welcome to MM207

Unit 4 Seminar

Binomial and the Discrete Probability Function (w/

Excel) 1

Definitions

• Statistical Experiment: Any process by which we obtain measurements or data.– In Unit 3 seminar we discussed dice. Rolling the dice is a

statistical experiment.• Random Variable: A random variable is the outcome of a

statistical experiment. We don’t know that this outcome will be before conducting the experiment– Discrete random variable: The possible values of the

experiment take on a countable number of results.• For the roll of a die there were 6 possible results.• Discrete variables have to be counted (like eggs)

– Continuous random variable: The possible values of the experiment are infinite.

• For example, measure the weight of 1 year old cows is continuous. The number of possibilities are uncountable. (500.12 pounds, 534.1534 pounds, etc…

• Continuous variables have to be measured (like weight or milk or gas)

Probability Distribution

• Probability distribution is the assignment of probabilities to specific values for a random variable or to a range of values for the random variable

• Plain English: Each random variable (outcome) from a random experiment has a particular probability of occurring.

• See page 196 Example 2

Score, x Probability, P(x)

1 0.16

2 0.22

3 0.28

4 0.20

5 0.14

Passive-Aggressive Traits

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5

Score

Probability distribution properties

• Mean: This is the expected value of a probability distribution. This is the outcome about which the distribution is centered.

μ = ∑ x P(x)

• Standard deviation: This is the spread of the data around the expected value (mean)

σ = √ ∑ (x – μ)2 P(x)

• How do you use the equations? Let’s use EXCEL!!!!.

Excel Procedure for Mean & Standard Deviation

(Try It Yourself 5 & 6; pp. 198-199)

1 2 3 4 5 6

X P(X) X*P(X) X - MEAN (X-MEAN)^2 P(X)*(X-MEAN)^2

0 0.16 0 -2.6 6.76 1.0816

1 0.19 0.19 -1.6 2.56 0.4864

2 0.15 0.3 -0.6 0.36 0.054

3 0.21 0.63 0.4 0.16 0.0336

4 0.09 0.36 1.4 1.96 0.1764

5 0.1 0.5 2.4 5.76 0.576

6 0.08 0.48 3.4 11.56 0.9248

7 0.02 0.14 4.4 19.36 0.3872

  1 2.6 3.72

    MEAN VARIANCE

   

    1.93

    STANDARD DEVIATION

Features of the Binomial Experiment• Fixed number of trials denoted by n• n trials are independent and performed under

identical conditions• Each trial has only two outcomes: success

denoted by S and failure denoted by F• For each trial the probability of success is the

same and denoted by p. The probability of failure is denote by q and q = 1 - p)

• The central problem is to determine the probability of x successes out of n trials. P(x) = ?

Example

n = 10p = 0.4x = 6

Find P(x = 6)

Using the binomial table (Table 2 A8-A10)Using the binomial formulaUsing Excel

Using the Binomial Formula

P(x) = nCx px qn-x

x = number of successesn = number of trialsp = probability of one successq = probability of one failure (1 – p)

nCx is the binomial coefficient give by nCx = n! / [x! (n-x)!]

Remember 4! = 4*3*2*1 = 24 and is called factorial notation.

Using the Binomial Formulan = 10p = 0.4x = 6

Find P(x = 6)

P(x) = nCx px qn-x

nCx = 10C6 = 210

px = 0.46 = 0.004096qn-x = 0.610-6 = 0.64 =0.1296

210* 0.004096 * 0.1296 = 0.111476736 ≈ 0.111

Using Excel to Calculate Binomial Probabilities

Using Exceln = 10p = 0.4x = 6

Find P(x = 6)

Click on the cell where you want the answer.Under fx, find BINOMDIST

Number_s: Enter 6Trials: Enter 10Probability: Enter 0.4Cumulative: FalseP(x = 6) = 0.111476736 ≈ 0.111

Finding Cumulative Probabilitiesn = 10p = 0.4x ≤ 6

Find P(x ≤ 6)

Find each probability using the binomial table or the formula.

P(x ≤ 6) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6)

= 0.006 + 0.040 + 0.121 + 0.215 + 0.251 + 0.201 + 0.111 = 0.945

Use the complementP(x ≤ 6) = 1 – P(x > 6) = 1 – [P(x = 7) + P(x = 8) + P(x = 9) + P(x = 10)]= 1 – [0.042 + 0.011 + 0.002 + 0.000] = 1 – 0.055 = 0.945

Finding Cumulative Probabilities con’tn = 10p = 0.4x ≤ 6

Find P(x ≤ 6)

Use Excel• Number_s: Enter 6• Trials: Enter 10• Probability: Enter .4• Cumulative: TrueP(x ≤ 6) = 0.945238118 ≈ 0.945

Mean and Standard Deviation of the binomial probability distribution

• Mean or expected number of successμ = np

• Standard deviationσ = √ npq

• Where:n = number of trialsp = probability of successq = probability of failure (q = 1 – p)

Computing the Mean, Standard Deviation, and Variance for a Binomial Distributionn = 10p = 0.4

Meanμ = npμ = 10 * 0.4μ = 4

Standard deviation

σ = √ npq

σ = √ 10 * 0.4 * 0.6

σ = √ 2.4 ≈ 1.549

Varianceσ2 = npq = 2.4

Have a great week!