Find the largest sample meeting a strict probability bound

The probability that each selected bulb is faulty stays constant at 0.01, and whether a bulb is faulty is independent of whether the other bulbs are faulty. As the Teacher note emphasizes, describe independence of events or trials, not “independent probability”. When the needed binomial conditions are not supplied, make the modelling assumptions explicit.

Let X count faulty bulbs among n, with $X\sim B(n,0.01)$. Then $P(X=0)>0.5$ gives $\binom n0(0.01)^0(0.99)^n>0.5$, hence $0.99^n>0.5$. Taking logs gives $\ln(0.99^n)>\ln0.5$, so $n\ln0.99>\ln0.5$. Because $\ln0.99<0$, reverse the inequality when dividing: $n<\ln0.5/\ln0.99=68.9676\ldots$. The largest integer n is 68. The sign reversal is the distinct Teacher warning.

1. Press Y=. 2. Press 2nd, VARS, select A:binompdf(, then ENTER. 3. Set trials:X, p:0.01, x value:0, choose Paste, then ENTER. The Y-editor contains Y1=binompdf(X,0.01,0). Here the calculator’s X input stands for the varying trial count n. 4. Press 2nd, GRAPH and scroll to the first probability below 0.5. Present the source boundary table, as the Teacher note specifically requires:

At 68 the probability exceeds 0.5; at 69 it does not. Thus the largest sample is 68. For a fixed p=0.01, the no-fault probability 0.99ⁿ decreases with n, so this boundary comparison establishes the largest value.

The source continuation explains TblStart (starting value) and ΔTbl (row increment). Press 2nd, WINDOW, set TblStart=0, ΔTbl=10, with independent/dependent entries on Auto. Press 2nd, GRAPH. The table brackets the crossing between n=60 and n=70. The screenshot shows ERROR at n=0 in this calculator command; this is a calculator-domain limitation, not a claim that the mathematical zero-trial distribution is invalid. Next set TblStart=60, ΔTbl=1 through 2nd, WINDOW, then 2nd, GRAPH to locate the exact integer boundary. The full coarse and refined source tables are retained in the diagram.