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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