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Worked solution

Allow an interval endpoint on either side of the fixed bound

X is normally distributed with mean 10 and standard deviation 2.

Find P(9<X<13)P(9<X<13) and P(X<12)P(X<12).

Find the possible values of a if the probability that X lies between a and 12 is 0.15.

If three values are taken at random, find the probability that two are less than 12 and the other greater than 12.

CJC · Discussion · Question 6

Answer

P(9<X<13)≈0.625P(9<X<13)\approx0.625, P(X<12)≈0.841P(X<12)\approx0.841; a≈11.0a\approx11.0 or 14.8; final probability 0.337 (3 s.f.).

Calculate the two initial probabilities

X∼N(10,22)X\sim N(10,2^2). The source gives P(9<X<13)=0.62466≈0.625P(9<X<13)=0.62466\approx0.625 and P(X<12)=0.84134≈0.841P(X<12)=0.84134\approx0.841, both rounded to 3 significant figures.

How did this part go?

Case 1: a below 12

If a<12, P(a<X<12)=0.15P(a<X<12)=0.15, so P(X<12)−P(X<a)=0.15P(X<12)-P(X<a)=0.15. Therefore P(X<a)=0.84134−0.15=0.69134P(X<a)=0.84134-0.15=0.69134, giving a≈11.0.

How did this part go?

Case 2: a above 12

If a>12, P(12<X<a)=0.15P(12<X<a)=0.15, so P(X<a)−P(X<12)=0.15P(X<a)-P(X<12)=0.15. Thus P(X<a)=0.15+0.84134=0.99134P(X<a)=0.15+0.84134=0.99134, giving a≈14.8. Both cases are possible because the upper tail beyond 12 exceeds 0.15.

How did this part go?

Count exactly two observations below 12

The source treats the three randomly taken values as independent observations. Exactly two below 12 and one above 12 has probability [P(X<12)]2P(X>12)(3!/2!)≈0.337[P(X<12)]^2P(X>12)(3!/2!)\approx0.337 to 3 significant figures. Use the unrounded cumulative probability in the calculation; the factor 3 chooses the position of the value above 12.

How did this part go?

Common mistakes

Keep both endpoint cases. For repeated observations, the ordering factor is needed for exactly two below 12.

Solution notes

Every ask, both source initial decimal/rounded results, both endpoint cases with each cumulative equation, and final repeated-value formula/order factor retained. Source independence assumption explicit; no marks printed.

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