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Worked solution

Compare two runners and distinguish a sum from a multiple

The times taken by two runners A and B to run 400 metre races are independent and normally distributed with means 45.0 seconds and 45.2 seconds and standard deviation 0.5 seconds and 0.8 seconds respectively. The two runners are to compete in a 400 metre race for which there is a track record of 44.5 seconds.

(i) Calculate the probability of runner A breaking the track record. Hence, show that the probability of runner B breaking the record is greater than that of runner A. marks

(ii) Find the probability of runner A beating runner B. marks

(iii) Find the probability that the total sum of four randomly chosen times of A is more than 4 times a randomly chosen time of B. marks

[2009/TPJC/II/11]

CJC · Review problems · Question 1

Answer

(i) A:0.159, B:0.191; (ii)0.584; (iii)0.406.

Define independent normal variables

Let X∼N(45,0.52)X\sim N(45,0.5^2) and Y∼N(45.2,0.82)Y\sim N(45.2,0.8^2) denote times in seconds. Smaller times are faster.

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(i) Break the record

P(X<44.5)=Φ(−1)≈0.159P(X<44.5)=\Phi(-1)\approx0.159 and P(Y<44.5)=Φ(−0.875)≈0.191P(Y<44.5)=\Phi(-0.875)\approx0.191 (3 s.f.). Since 0.191>0.159, B has the greater record-breaking probability despite the slower mean.

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(ii) Compare one race each

Independence gives X−Y∼N(−0.2,0.52+0.82)=N(−0.2,0.89)X-Y\sim N(-0.2,0.5^2+0.8^2)=N(-0.2,0.89). Hence P(X<Y)=P(X−Y<0)≈0.584P(X<Y)=P(X-Y<0)\approx0.584 (3 s.f.).

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(iii) Compare a four-time sum with one scaled time

For four independently chosen A-times and one independent B-time, put D=X1+X2+X3+X4−4YD=X_1+X_2+X_3+X_4-4Y. Then

E(D)=4(45)−4(45.2)=−0.8,E(D)=4(45)-4(45.2)=-0.8,

Var⁡(D)=4(0.52)+42(0.82)=11.24.\begin{aligned}&\operatorname{Var}(D)\\ & =4(0.5^2)+4^2(0.8^2)\\ & =11.24.\end{aligned}

Thus D∼N(−0.8,11.24)D\sim N(-0.8,11.24) and P(D>0)≈0.406P(D>0)\approx0.406 (3 s.f.).

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Common mistakes

Winning means a shorter time; record-breaking is a lower-tail event.

Solution notes

All source parts and moments retained. Independence across the four randomly chosen A-times, used by the printed solution, is explicit.

Read the hint