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Worked solution

Practise equality, tails and a centred absolute inequality

For X∼B(23,0.34)X\sim B(23,0.34), evaluate (i) P(X=4)P(X=4), (ii) P(X≤6)P(X\le6), (iii) P(X<5)P(X<5), (iv) P(X>4)P(X>4), (v) P(X≥14.5)P(X\ge14.5), (vi) P(7<X≤16)P(7<X\le16), and (vii) P(∣X−8∣<5.6)P(\lvert X-8\rvert<5.6).

CJC · Lecture · Self-Practice 1

(i) Equality

binompdf(23,0.34,4) gives the printed answer P(X=4)=0.0441P(X=4)=0.0441 to 3 significant figures.

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(ii) Inclusive lower tail

binomcdf(23,0.34,6) gives P(X≤6)=0.287P(X\le6)=0.287 to 3 significant figures.

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(iii) Strict lower tail

Because X is an integer, P(X<5)=P(X≤4)P(X<5)=P(X\le4). Use binomcdf(23,0.34,4) to obtain 0.0669 to 3 significant figures.

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(iv) Strict upper tail

P(X>4)=1−P(X≤4)P(X>4)=1-P(X\le4). Use 1-binomcdf(23,0.34,4) to obtain 0.933 to 3 significant figures.

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(v) Non-integer threshold

P(X≥14.5)=P(X≥15)=1−P(X≤14)P(X\ge14.5)=P(X\ge15)=1-P(X\le14). Use 1-binomcdf(23,0.34,14) to obtain 0.00219 to 3 significant figures.

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(vi) Mixed endpoints

P(7<X≤16)=P(X≤16)−P(X≤7)P(7<X\le16)=P(X\le16)-P(X\le7). Use binomcdf(23,0.34,16)-binomcdf(23,0.34,7) to obtain 0.546 to 3 significant figures.

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(vii) Split the absolute value

∣X−8∣<5.6\lvert X-8\rvert<5.6 means −5.6<X−8<5.6-5.6<X-8<5.6, hence 2.4<X<13.62.4<X<13.6. The integer values are 3 through 13. Thus P(3≤X≤13)=P(X≤13)−P(X≤2)=0.987P(3\le X\le13)=P(X\le13)-P(X\le2)=0.987 to 3 significant figures.

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Common mistakes

Use integer support to resolve strict, non-integer and absolute-value bounds.

Solution notes

The source prints these seven answers without intermediate working. The event conversions and calculator expressions are authored teaching steps; all seven printed asks and rounded results are retained.

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