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Worked solution

Find the smallest battery sample below a probability bound

Twenty percent of batteries produced by a machine are faulty. Find the smallest sample size for which the probability of fewer than 12 faulty batteries is less than 0.65.

CJC · Lecture · Example 5

Define the model and integer event

Let X count faulty batteries among n sampled batteries. Under the source constant-probability independent-battery model, X∼B(n,0.2)X\sim B(n,0.2). Require P(X<12)<0.65P(X<12)<0.65, equivalently P(X≤11)<0.65P(X\le11)<0.65. These count, parameter and inequality entries complete the Student blanks using the Teacher copy.

How did this part go?

Use the cumulative calculator function

Enter Y1=binomcdf(X,0.2,11) and press 2nd, GRAPH. Here the calculator X is the sample-size input n. The source instruction calls this a probability distribution function P(X=x)P(X=x), but its actual screen and required event use the cumulative probability P(X≤11)P(X\le11). Retain the Teacher boundary table:

nP(X≤11)P(X\le11)
520.65932
530.63301
540.60645
How did this part go?

Select the first qualifying integer

At n=52 the probability 0.65932 is above 0.65; at n=53 the probability 0.63301 is below 0.65, and n=54 gives 0.60645. Thus the smallest sample size is 53. The cumulative probability of at most 11 faults cannot increase as independent trials with the same positive fault probability are added, so checking this first crossing establishes the minimum.

How did this part go?

Compare your sketch

Find the first sample size below a probability bound

  1. 1. Translate the strict count bound

    Fewer than 12 faulty batteries means at most 11. Keep the fault probability 0.2 fixed and vary the sample size n.

  2. 2. Enter the cumulative probability

    In Y= enter the following, then press 2nd → GRAPH. The calculator X stands for n.

    Y1
    binomcdf(X,0.2,11)

  3. 3. Check both sides of the first crossing

    Compare the cumulative probability with 0.65.

    Check both sides of the first crossing
    nDisplayed P(at most 11)
    520.65932
    530.63301
    540.60645

    Smallest qualifying integer: n = 53

At 52 the probability is above 0.65; at 53 it is below. This cumulative probability cannot increase when independent trials with the same positive fault probability are added. The source calls the command a point probability in one instruction; its actual binomcdf entry is cumulative.

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Two reconstructed screens retain Y1=binomcdf(X,.2,11), all other Y entries blank and the full 50–60 table. The varied calculator X is sample size n. Printed labels calling this a point probability are qualified in the working; the actual command is cumulative.
Two reconstructed screens retain Y1=binomcdf(X,.2,11), all other Y entries blank and the full 50–60 table. The varied calculator X is sample size n. Printed labels calling this a point probability are qualified in the working; the actual command is cumulative.
Two reconstructed screens retain Y1=binomcdf(X,.2,11), all other Y entries blank and the full 50–60 table. The varied calculator X is sample size n. Printed labels calling this a point probability are qualified in the working; the actual command is cumulative.
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Common mistakes

A strict “fewer than 12” event uses the cumulative probability at 11.

Read the hint