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Worked solution

Find a binomial game limit and a daily-average probability

At a fun-fair stall, the probability of winning a prize in each game is 0.05.

(a) Find the largest number of games in a day if the probability of more than 5 prizes is to be less than 0.1. marks

(b) The stall opens for 70 days, with 60 games each day. Find the probability that the average number of prizes per day is between 2.5 and 3 inclusive. marks

[2010/TJC/II/10(a)&(c)]

CJC · Review problems · Question 5

Answer

(a) Maximum 63 games. (b) Printed CLT 0.4930.493; exact binomial 0.5133325709…0.5133325709\ldots.

(a) Compare integer binomial tails

For nn independent games, X∼B(n,0.05)X\sim B(n,0.05). Require 1−P(X≤5)<0.11-P(X\le5)<0.1. The source table is:

nnP(X>5)P(X>5)
620.08906
630.09449
640.1001

The maximum is 63. Adding a game cannot decrease the number of prizes, so once the bound fails at 64 it cannot recover for larger nn.

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(b) Printed CLT average

The daily count is Y∼B(60,0.05)Y\sim B(60,0.05), with mean 3 and variance 2.85. Thus Yˉ∼˙N(3,2.85/70)\bar Y\mathrel{\dot\sim}N(3,2.85/70) across 70 independent days, giving P(2.5≤Yˉ≤3)≈0.493P(2.5\le\bar Y\le3)\approx0.493, to 3 s.f.

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(b) Authored exact check including both endpoints

The total is exactly T∼B(4200,0.05)T\sim B(4200,0.05). The inclusive mean event is 175≤T≤210175\le T\le210. The exact probability is 0.5133325709… . The printed 0.493 is an unadjusted continuous CLT approximation, not this exact inclusive-binomial result.

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Common mistakes

For an inclusive discrete interval, retain the endpoint masses when comparing with an exact total distribution.

Read the hint