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Worked solution

Respect the first-miss stopping rule before comparing rounds

A fairground game involves shooting at a moving target. A round has a maximum of 3 shots and ends immediately after a miss. Each hit scores 10 points. Linda hits with probability 0.6 on each shot, and all shots taken are independent.

(i) Find the probability that she scores 30 points in a round. marks

X is her score in one round.

(ii) Find its distribution. marks

(iii) Find its mean and variance. marks

(iv) A game consists of two rounds. Find the probability that she scores more in round 2 than round 1. marks

[2017/SAJC/II/8]

CJC · Review problems · Question 8

Answer

(i)0.216; (ii)distribution above; (iii)mean 11.76, variance 137.7024; (iv)0.357504≈0.358 (3 s.f.).

(i) Three consecutive hits

P(X=30)=P(hit,hit,hit)=0.63=27/125=0.216P(X=30)=P(\text{hit,hit,hit})=0.6^3=27/125=0.216.

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(ii) Retain all stopping paths and distribution entries

xP(X=x)P(X=x)
00.40.4
100.6(0.4)=0.240.6(0.4)=0.24
200.62(0.4)=0.1440.6^2(0.4)=0.144
300.63=0.2160.6^3=0.216

For scores 0,10,20, the final shot is a miss. After three hits the round ends without a fourth shot.

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(iii) Retain all first and second moment terms

E(X)=0(0.4)+10(0.24)+20(0.144)+30(0.216)=11.76E(X)=0(0.4)+10(0.24)+20(0.144)+30(0.216)=11.76. E(X2)=02(0.4)+102(0.24)+202(0.144)+302(0.216)=276E(X^2)=0^2(0.4)+10^2(0.24)+20^2(0.144)+30^2(0.216)=276. Therefore Var⁡(X)=276−(11.76)2=137.7024\operatorname{Var}(X)=276-(11.76)^2=137.7024.

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(iv) Retain the three disjoint comparisons

Let X1,X2X_1,X_2 be the respective round scores. The event is (X1=0,X2≥10)(X_1=0,X_2\ge10) or (X1=10,X2≥20)(X_1=10,X_2\ge20) or (X1=20,X2=30)(X_1=20,X_2=30). Independent shots across both rounds give P(X2>X1)=P(X1=0)P(X2≥10)+P(X1=10)P(X2≥20)+P(X1=20)P(X2=30)=0.4(1−0.4)+0.24(0.144+0.216)+0.144(0.216)=0.357504=0.358P(X_2>X_1)=P(X_1=0)P(X_2\ge10)+P(X_1=10)P(X_2\ge20)+P(X_1=20)P(X_2=30)=0.4(1-0.4)+0.24(0.144+0.216)+0.144(0.216)=0.357504=0.358 to 3 significant figures.

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Common mistakes

This score is not 10 times a binomial count of three shots: a miss prevents later shots.

Solution notes

All four asks/marks, complete source distribution with products, all moment terms and each two-round comparison/product retained.

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