exercise:7d7451399d

Jan 16'24

Exercise

A group of 100 people start a Scissor Usage Support Group. The rate at which members enter and leave the group is dependent on whether they are right-handed or left-handed.

You are given the following:

(i) The initial membership is made up of 75% left-handed members (L) and 25% right-handed members [math](\mathrm{R})[/math]

(ii) After the group initially forms, 35 new (L) and 15 new (R) join the group at the start of each subsequent year

(iii) Members leave the group only at the end of each year

(iv) [math]q^{L}=0.25[/math] for all years

(v) [math]q^{R}=0.50[/math] for all years

Calculate the proportion of the Scissor Usage Support Group's expected membership that is left-handed at the start of the group's [math]6^{\text {th }}[/math] year, before any new members join for that year.

0.76
0.81
0.86
0.91
0.96

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AAdmin

Jan 16'24

Answer: C

The number of left-handed members at the end of each year [math]k[/math] is:

[math]L_{0}=75[/math] and [math]L_{1}=(75)(0.75)[/math]

Thereafter, [math]L_{k}=L_{k-1} \times 0.75+35 \times 0.75=75 \times 0.75^{k}+35 \times\left(0.75+0.75^{2}+\ldots 0.75^{k-1}\right)[/math]

Similarly, the number of right-handed members after each year [math]k[/math] is:

[math]R_{0}=25[/math] and [math]R_{1}=(25)(0.5)[/math]

Thereafter, [math]R_{k}=R_{k-1} \times 0.50+15 \times 0.50=25 \times 0.50^{k}+15 \times\left(0.50+0.50^{2}+\ldots 0.50^{k-1}\right)[/math]

At the end of year 5, the number of left-handed members is expected to be 89.5752, and the number of right-handed members is expected to be 14.8435 .

The proportion of left-handed members at the end of year 5 is therefore

[math]\frac{89.5752}{89.5752+14.8438}=0.8578[/math]