Worked solution · Exam P

SOA Exam P Sample Question #116 Solution

This Exam P sample reference tests Normal Approximation to Binomial. For Binomial(40,0.5), the continuity-corrected tails at cutoffs 23 and 24 are approximately 0.134 and 0.077. The 0.10 crossing therefore occurs at N=23, choice A.

Independent analysis by ActuaryProof · reviewed October 6, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

Read the solution Open official source

Independent solution

How to solve this Normal Approximation to Binomial question

Setup

The binomial count has mean 20 and variance 10. The requested integer is the point at which the approximated upper tail first passes from above 0.10 to below 0.10.

\mu=20,\quad \sigma=\sqrt{10}

Model

Apply a continuity correction to the event above 23 by using the boundary 23.5. Its normal upper-tail approximation is 0.134.

\Pr(C>23)\approx\Pr\left(Z>\frac{23.5-20}{\sqrt{10}}\right)=0.134

Compute

The next boundary, 24.5, gives an upper tail of 0.077. Since 0.10 lies between these adjacent tails, the required integer is 23.

\Pr(C>24)\approx\Pr\left(Z>\frac{24.5-20}{\sqrt{10}}\right)=0.077

Answer

The adjacent-tail condition selects N=23.

\boxed{23\quad\text{(A)}}

Test the method on unseen problems

Probability Univariate Distributions Practice Workbook

30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.

See the cover and sample pages

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

BAt cutoff 25 the upper tail is already below 0.10, so it is beyond the adjacent crossing established at 23 and 24.
CA cutoff of 32 is far above the mean and cannot satisfy the required above-0.10 first inequality.
DA cutoff of 33 is also far beyond the first crossing below 0.10.
EA cutoff of 35 is still farther into the upper tail and is not the first qualifying boundary.

Original practice · fully worked

Original variant: normal approximation for a quality score

A quality test has 100 independent items, each passed with probability 0.40. Find the integer N for which the normal approximation with continuity correction gives P(X>N)>0.05 but P(X>N+1)<0.05.

A
45
B
46
C
47
D
48
E
49

Answer and calculation

The mean is 40 and standard deviation √(24). Corrected cutoffs 47.5 and 48.5 give upper tails about 0.063 and 0.042, so N=47, choice C.

Setup

The count has mean 40 and variance 24. The target integer is located by comparing two adjacent continuity-corrected upper tails with 0.05.

\mu=40,\quad \sigma=\sqrt{24}

Model

The event above 47 uses boundary 47.5 and has approximated upper tail 0.063.

\Pr(X>47)\approx\Pr(Z>1.531)=0.063

Compute

The event above 48 uses boundary 48.5 and has approximated upper tail 0.0415. The crossing therefore occurs at 47.

\Pr(X>48)\approx\Pr(Z>1.735)=0.0415

Answer

The probability crosses 0.05 between the two tails at N=47.

\boxed{47\quad\text{(C)}}

Keep practicing

Next steps for Exam P Univariate Random Variables

Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.

More Exam P Univariate Random Variables practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Normal Approximation to Binomial question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: normal approximation for a quality score

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables