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Binomial Settings

Learn binomial settings, binomial probabilities.

Lesson 3 of 32 parts Practice Flashcards

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Binomial Settings

Objective: Explain binomial settings, apply the stated method, and check its conditions.

A binomial model counts successes in a fixed number n of independent trials with two outcomes and constant success probability p. Its mean is np and variance is np(1−p). Check all conditions; sampling without replacement may violate independence. A fixed trial count, two outcomes, independence, and constant p are required.

Formulas

$E[X]=np; Var(X)=np(1−p)$

Worked examples

Try itFor a binomial variable with n=42 and p=1/6, find its mean.Show answer

Step 1: A binomial mean is np. Step 2: Compute (42)(1/6). Step 3: The mean is 7; binomial assumptions include independent trials and constant p.

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1

Binomial Settings

Objective: Explain binomial settings, apply the stated method, and check its conditions.

A binomial model counts successes in a fixed number n of independent trials with two outcomes and constant success probability p. Its mean is np and variance is np(1−p). Check all conditions; sampling without replacement may violate independence. A fixed trial count, two outcomes, independence, and constant p are required.

Formulas

$E[X]=np; Var(X)=np(1−p)$

Worked examples

Try itFor a binomial variable with n=42 and p=1/6, find its mean.Show answer

Step 1: A binomial mean is np. Step 2: Compute (42)(1/6). Step 3: The mean is 7; binomial assumptions include independent trials and constant p.

2

Binomial Probabilities

Objective: Explain binomial probabilities, apply the stated method, and check its conditions.

For X~Binomial(n,p), P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ. The combination factor counts different placements of successes. For k=0, every trial fails and the probability is (1−p)ⁿ. All model assumptions remain necessary. The combinatorial factor is needed except in special endpoint cases.

Formulas

$P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ$

Worked examples

Try itFor X~Binomial(7,1/2), find P(X=0).Show answer

Step 1: All trials must fail. Step 2: Each failure has probability 1/2, independently. Step 3: P(X=0)=(1/2)^7=1/128.

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