2000 Paper 3 Q13

Year: 2000
Paper: 3
Question Number: 13

Course: UFM Statistics
Section: Probability Generating Functions

Difficulty: 1700.0 Banger: 1516.0

probability generating functions PGF dice geometric distribution expectation order statistics first success last success inclusion-exclusion

Problem

A set of $n$ dice is rolled repeatedly. For each die the probability of showing a six is $p$. Show that the probability that the first of the dice to show a six does so on the $r$th roll is $$q^{n r } ( q^{-n} - 1 )$$ where $q = 1 - p$. Determine, and simplify, an expression for the probability generating function for this distribution, in terms of $q$ and $n$. The first of the dice to show a six does so on the $R$th roll. Find the expected value of $R$ and show that, in the case $n = 2$, $p=1/6$, this value is $36/11$. Show that the probability that the last of the dice to show a six does so on the $r$th roll is \[ \big(1-q^r\big)^n-\big(1-q^{r-1}\big)^n. \] Find, for the case $n = 2$, the probability generating function. The last of the dice to show a six does so on the $S$th roll. Find the expected value of $S$ and evaluate this when $p=1/6$.

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1516.0

Banger Comparisons: 1

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Problem source

A set of $n$ dice  is rolled repeatedly. 
For each die the probability  
of showing a six is $p$. Show that  
the probability that the first of the dice to show a six  
does so on the $r$th roll is 
$$q^{n  r } ( q^{-n} - 1 )$$ 
where $q = 1 - p$. 
 
Determine, and simplify,  
an expression for the probability generating function  
for this distribution, in terms of $q$ and $n$. 
The first of the dice to show a six does so on the $R$th roll. Find the  
expected value  of $R$ and show that, in the case 
 $n = 2$, $p=1/6$, this value  is $36/11$. 
 
Show that  the probability that the last of the dice to show a  
six does so on the $r$th roll 
is 
\[ 
\big(1-q^r\big)^n-\big(1-q^{r-1}\big)^n. 
\] 
Find, for the  
case $n = 2$, the probability generating function. The  
last of the dice to show a six does so on the $S$th roll. Find the expected value 
of $S$ and  evaluate this when $p=1/6$.

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