Porisma

Worked solution

Find the least integer threshold for late days

A student has probability 0.1 of being late for school on any day. In 18 days, the probability of being late on fewer than r days must exceed 0.85. Find the least r.

CJC · Lecture · Example 6

Name the count and translate the requirement

Let X count late days among 18 days. With the source independent-day model and constant probability 0.1, X∼B(18,0.1)X\sim B(18,0.1). The condition is P(X<r)>0.85P(X<r)>0.85, hence P(X≤r−1)>0.85P(X\le r-1)>0.85 for integer r. The source treats r as an integer day-count threshold. Without that interpretation there would be no least real r: every r just above 3 would already include X=3.

How did this part go?

Use the source threshold table

Enter Y1=binomcdf(18,0.1,X-1), then press 2nd, GRAPH. The calculator X now stands for r. Retain the Teacher completion:

rP(X≤r−1)P(X\le r-1)
30.7338
40.9018
50.97181

The probability increases with r. At 3 it is below 0.85; at 4 it is above 0.85. Therefore the least integer r is 4.

How did this part go?

Compare your sketch

Find the least integer threshold

  1. 1. Identify the unknown

    Keep n=18 and p=0.1 fixed. For an integer threshold r, fewer than r late days means at most r−1.

  2. 2. Enter a varying threshold

    In Y= enter the command below, then press 2nd → GRAPH. Here the calculator X stands for r, not n.

    Y1
    binomcdf(18,0.1,X-1)

  3. 3. Find the first value above 0.85

    Check adjacent integer thresholds.

    Find the first value above 0.85
    rDisplayed P(fewer than r)
    30.7338
    40.9018
    50.97181

    Least integer threshold: r = 4

The threshold is an integer in this question. Keep the r−1 adjustment and the strict probability bound: the entry at r=3 is too small, while r=4 qualifies.

View original calculator screens
Expand diagram

Use the scrollbars or arrow keys to inspect details.

Two reconstructed screens retain Y1=binomcdf(18,.1,X−1), blank other Y entries and the full 1–11 table. The calculator X is threshold r, so its cumulative argument is r−1.
Two reconstructed screens retain Y1=binomcdf(18,.1,X−1), blank other Y entries and the full 1–11 table. The calculator X is threshold r, so its cumulative argument is r−1.
Two reconstructed screens retain Y1=binomcdf(18,.1,X−1), blank other Y entries and the full 1–11 table. The calculator X is threshold r, so its cumulative argument is r−1.
Open original screens (new tab)

Common mistakes

Vary the threshold r, keeping n=18 fixed; use r−1 for a strict count bound.

Read the hint