Independent solution

How to solve this Binomial Distribution question

Setup

Setup

Use a success indicator for an item that is not defective. Its success probability is the complement of the defect probability.

q=1−p,X∼Binomial⁡(50,q)q=1-p,\qquad X\sim\operatorname{Binomial}(50,q)

Model

Model

For a count k, choose the k successful positions and attach the appropriate success and failure factors.

Pr⁡(X=k)=(50k)(1−p)kp50−k\Pr(X=k)=\binom{50}{k}(1-p)^k p^{50-k}

Compute

Compute

At least n means that the count ranges from n through the maximum possible value 50.

Pr⁡(X≥n)=∑k=n50(50k)(1−p)kp50−k\Pr(X\ge n)=\sum_{k=n}^{50}\binom{50}{k}(1-p)^k p^{50-k}

Answer

Answer

The required inequality uses that upper-tail expression.

∑k=n50(50k)(1−p)kp50−k≥0.95(A)\boxed{\sum_{k=n}^{50}\binom{50}{k}(1-p)^k p^{50-k}\ge0.95\quad\text{(A)}}