Copyright [2020] [Alexander Craig Murph] Licensed under the
	Educational Community License, Version 2.0 (the "License"); you may
	not use this file except in compliance with the License. You may
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####### QUESTION AREA

<eqn>
sub NBcumulativeprob {
my ($r, $p, $k) = @_;
$total_prob = 0;
for( $a = $r; $a <= $k; $a = $a + 1 ) {
   $first = $a - 1;
   $second = $r - 1;
   $probability = combination($first,$second)*($p**$r)*((1-$p)**($a-$r));
   $total_prob = $total_prob + $probability;
}
return $total_prob;
}
" "
</eqn>

<eqn>
# Stuff for binomial question
$prob = randnum(35, 80, 1)/100;
$binom_n = randnum(8,15,1);
$temp_n = $binom_n - 1;
$binom_x = randnum(5,$temp_n,1);
$bin_pdf_answer = distr("pdf=bin, n=$binom_n, p=$prob, x=$binom_x");
$prob_comp = 1 - $prob;
$bin_pdfcomplement_answer = distr("pdf=bin, n=$binom_n, p=$prob_comp, x=$binom_x");
$bin_cdf_answer = 1 - distr("cdf=bin, n=$binom_n, p=$prob, x=$binom_x");

# Stuff for negative binomial
$nb_k = randnum(7, 25, 1);
$temp_nb_k = $nb_k -2;
$nb_r = randnum(2, $temp_nb_k, 1);
$first = $nb_k - 1;
$second = $nb_r - 1;
$equal_to_k_prob = combination($first,$second)*($prob**$nb_r)*((1-$prob)**($nb_k-$nb_r));
$more_than_k_prob = 1 - NBcumulativeprob($nb_r, $prob, $nb_k);



$lambda = randnum(1,8,1);
$pois_x = randnum(3,8,1);
$less_than = $pois_x - 1;
$less_than_x = distr("cdf=poi, lambda=$lambda, x=$less_than");
$sd = sqrt($lambda);
" "
</eqn>

Schoolyard jeers can sting and bruise, but perhaps not as much as a 5-man tackle.  It is best to turn the cheek when face-to-face with an adolescent tormentor.  Unless, of course, that villain is holding the ball on the rugby field.  In that case...full speed ahead!!  
<br>
<br>
You've practiced all summer and you are ready to kick-butt this coming rugby season.  From all your summer skirmishes, you know that the probability that you win any game is <eqn $prob>.  Suppose the probability that you win a given game is independent of your victory (or lack thereof) in any other game, and that ties are not possible.
<br>
<br>
a) What's the probability that, during the next <eqn $binom_n> games, you win <eqn $binom_x> of them? <_>
<br>
<br>
b) What's the probability that, during the next <eqn $binom_n> games, you lose <eqn $binom_x> of them? <_>
<br>
<br>
c) What's the probability that, during the next <eqn $binom_n> games, you win more than <eqn $binom_x> of them? <_>
<br>
<br>
d) What's the probability that you have your <eqn $nb_r>th win on the <eqn $nb_k>th game you play? <_>
<br>
<br>
e) What's the probability that you have your <eqn $nb_r>th win sometime after the <eqn $nb_k>th game you play? <_>
<br>
<br>
<SECTION>f) All of the questions up until now are applications of various probability distributions we have discussed in this class.  What are the assumptions needed for <i>all</i> of these probability distributions? <_>
<br>
<br>
<SECTION>g) Say the average number of trys (the means of scoring points in Rugby) your team scores in an hour is <eqn $lambda>.  What is the probability that your team scores less than <eqn $pois_x> trys in the next hour? <_><br>
<br>
h) What is the standard deviation of trys in the next hour? <_>



####### ANSWER AREA


<EQN $bin_pdf_answer> {tab} 0.001
<EQN $bin_pdfcomplement_answer> {tab} 0.001
<EQN $bin_cdf_answer> {tab} 0.001
<EQN $equal_to_k_prob> {tab} 0.001
<EQN $more_than_k_prob> {tab} 0.001
<SECTION>The individual trials are independent, each trial has an equal probability of success, and each outcome is binary: only one of two possible things can happen.
The individual trials are independent, each trial has an equal probability of success, and there are a fixed total number of trials.
The trials must happen in a fixed unit of time, each trial has an equal probability of success, and each outcome is binary: only one of two possible things can happen.
The individual trials are independent, the trials must happen in a fixed unit of time, and each outcome is binary: only one of two possible things can happen.
The individual trials are independent, the trials must happen in a fixed unit of time, and there are a fixed total number of trials.
Each trial has an equal probability of success, the trials must happen in a fixed unit of time, and there are a fixed total number of trials.
<SECTION><EQN $less_than_x> {tab} 0.001
<EQN $sd> {tab} 0.001