Central Bucks School District
Ch. 17: Binomials
Example: Computer chips have a 25% chance of being defective. Create the probability distribution for X, if X is the # of defective chips in a sample of 3. What is the probability of having 2 or more defective chips?
|X |0 |1 |2 |3 |
|P(X) | | | | |
a) What is the probability of having 2 or more defective chips?
[pic]
b) What is the probability of having 1 or less defective chips?
[pic]
c) What is the probability of having exactly 2 defective chips?
[pic]
BINOMIAL MODELS:
• Interested in the number of successes in a set number of trials
• 4 conditions that must apply:
o Only 2 possible outcomes (success/failure)
o Probability of success remains constant (called p)
o Number of trials is set/known (called n)
o Independent trials
▪ 10% Condition: If we cannot assume independence, we can proceed as long as the sample is smaller than 10% of the population (pop [pic] 10n)
• If these 4 conditions apply, we have a Bernoulli trial
Notation: B(n, p)
probability of success
µX= n * p = E(x) σX= [pic]
(1-p)
Example: It is known that only 15% of the population is left handed. Create a probability distribution for the number of left handed people in a sample of 3.
|X |P(X) |
|0 |.6141 |
|1 |.3251 |
|2 |.0574 |
|3 | .0034 |
Quicker way to get probabilities:
Formula: P(X = k) = [pic]
combinations [pic]
|X |P(X) |
|0 |[pic] |
|1 |[pic] |
|2 | [pic] |
|3 | [pic] |
Example: I am playing a game in which I have a 39% chance of winning each time I play. Create the probability distribution for the number of wins out of 5 plays of the game.
STEP 1: Check if the problem is binomial
1. Two outcomes: Winning/Not Winning
2. P = .39 and remains constant (q = .61)
3. Number of trials is set at 5
4. Independence is not stated but we can assume independence since more than 50 games will be played.
STEP 2: Create the probability distribution
|X |P(X) |
|0 |[pic] |
|1 |[pic] |
|2 | [pic] |
|3 | [pic] |
|4 | [pic] |
|5 | [pic] |
STEP 3: answer questions
P(X=2) = .3452 P(X ................
................
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