You wander around the casino, and see this game called Keno. It's a bit like a lottery. There are 80 balls numbered 1 through 80, and they will pick 20 of them randomly. They have a catchy little flier about all the Keno bets you can make:

The whole flier is here.
Now, we're talking entertainment! You know there's no way that these tempting little bets are going to make you money in the long run, but have a little mathematical problem to solve, and that's better than gambling! Your job is to figure out which of these bets is the best  in other words, which one will lose you the least amount of money in the long run.
Let's analyze the bet sheet. On the "2Bit Menu" you will pay a quarter per game, which you will not get back. Let's say you choose the "5 catch win." This means that you will pick five numbers. They will pick 20. If exactly three of your five numbers are in their twenty, they pay you a quarter (i.e. you get your money back). If exactly four of your five are in their twenty, then they pay you a dollar. And if all five are your numbers are in their twenty, you win $200!!!!! Whoohoo!!!
So, suppose you want to calculate what the average return on your investment will be. The probability of matching exactly three balls is 0.0839351. The probability of matching exactly four balls is 0.0120923. And the probability of matching exactly five is 0.000644925. I'll show you how to calculate those later. So, to calculate your return:
In Keno, suppose you pick p balls. Then the number of ways to match exactly c of those p in the twenty randomly chosen balls is:
So, the probability of you matching exactly exactly c of p is:
Let's take a concrete example. If we choose five balls and want to match exactly three, that's: binom(75,17) * binom(5,3) / binom(80,20). Which equals:
Unfortunately, factorial grows so quickly that we cannot compute 75!, much less 17! directly (on my Mac the last factorial I can compute is 16!after that I get overflow).
Fortunately, we can cancel quite a few of these terms and make the multiplication manageable. For example, (75!)/(80!) = 1/(80*79*78*77*76). And (5!)/(3!) = 5*4. We can keep cancelling until we get:
This equals 0.0839351.
(For the true nerds, yes, we can do a prime factorization and cancel many more terms, but for the purposes of this lab, we'll just do simple cancellations).
If you are reeling a bit from this math, that is okay. You may have to reread this section a few times before you fully understand it. What is important is that you understand how to compute a binomial coefficient and an expected return. We have given you the formula for computing the probability of catching c out of p picks and you can simply apply that formula in this lab.
The first implements a class called a "product", defined in product.h:
#ifndef _PRODUCT_H #define _PRODUCT_H #include <iostream> #include <vector> #include <cstdlib> using namespace std; class Product { public: void Invert(); void Multiply_Number(int n); void Divide_Number(int n); void Multiply_Factorial(int n); void Divide_Factorial(int n); void Multiply_Binom(int n, int k); void Divide_Binom(int n, int k); void Clear(); void Print(); double Calculate_Product(); protected: vector <int> numerator; vector <int> denominator; }; #endif 
This represents a product as two vectors  a numerator and a denominator. Thinks of these two vectors as representing the factors in the product. Each entry in the vector represents the number of times the factor appears in the product. For example, the product:
2 * 3 * 3 * 5 * 5 * 5 * 6can be represented using the vector { 0, 0, 1, 2, 0, 3, 1}.
The numerator and denominator vectors will start as empty vectors. When you call Multiply_Number(i), you will make sure that numerator[i] and denominator[i] exist, and then you will increment numerator[i]. When you call Divide_Number(i), you will again make sure that numerator[i] and denominator[i] exist, and then you will increment denominator[i]. You don't have to error check, but you may assume that Multiply_Number(i) and Divide_Number(i) are always called on positive integers. So, if you make the following calls:
Product p; p.Multiply_Number(4); p.Multiply_Number(5); p.Multiply_Number(5); p.Divide_Number(5); 
Then p's numerator will be a vector containing: { 0, 0, 0, 0, 1, 2 } and p's denominator will be a vector containing: { 0, 0, 0, 0, 0, 1 }.
The other calls are defined as follows:
For example, the following code:
Product p; p.Multiply_Number(5); p.Multiply_Number(4); p.Divide_Number(3); p.Print(); 
Will print:
4 * 5 / 3 
And the following code:
Product p; p.Print(); p.Multiply_Binom(6,3); p.Print(); p.Multiply_Number(5); p.Divide_Number(4); 
Will print:
1 4 * 5 * 6 / 2 / 3 5 * 5 * 6 / 2 / 3 
To help you test, I have a program called prod_tester.cpp in the lab directory, which lets you enter commands on standard input to manipulate a product:
UNIX> prod_tester Help? Bad command. Should be one of CLEAR, CALCULATE, INVERT, PRINT, MULTIPLY, DIVIDE, MULT_EXP, DIV_EXP, MULT_BINOM or DIV_BINOM. PRINT 1 MULTIPLY 5 PRINT 5 DIVIDE 5 PRINT 1 MULT_BINOM 75 17 MULT_BINOM 5 3 DIV_BINOM 80 20 PRINT 4 * 5 * 18 * 19 * 20 * 59 * 60 / 2 / 76 / 77 / 78 / 79 / 80 CALCULATE 0.0839351 <CNTLD> UNIX>When you compile your product.cpp with prod_tester.cpp, your output should match mine exactly. Many of the grading script's tests will use prod_tester to test your program.
So, for example, the "5 catch win" above would be represented with:
0.25 5 3 0.25 4 1 5 200Now the first line of your program should print out the bet, and the second line should print the balls picked. The bet should be padded to two decimal places (use printf instead of cout). Then for each payout, you should print the probability of winning and the expected return (probability times payout). Use cout for these lines and have them match mine exactly. Finally, you should print the expected return per bet, which is the sum of all expected returns minus the bet. Have this padded to two decimal places (use printf instead of cout). Finally, print the normalized return, which is the expected return divided by the bet. Again, pad that to two decimal places.
This lets us evaluate all the Keno options in the flier. First, the "2bit menu":
UNIX> echo "0.25 5 3 0.25 4 1 5 200"  keno Bet: 0.25 Balls Picked: 5 Probability of catching 3 of 5: 0.0839351  Expected return: 0.0209838 Probability of catching 4 of 5: 0.0120923  Expected return: 0.0120923 Probability of catching 5 of 5: 0.000644925  Expected return: 0.128985 Your return per bet: 0.09 Normalized: 0.35 UNIX> echo "0.25 6 3 .25 4 .50 5 10 6 500"  keno Bet: 0.25 Balls Picked: 6 Probability of catching 3 of 6: 0.12982  Expected return: 0.0324549 Probability of catching 4 of 6: 0.0285379  Expected return: 0.014269 Probability of catching 5 of 6: 0.00309564  Expected return: 0.0309564 Probability of catching 6 of 6: 0.000128985  Expected return: 0.0644925 Your return per bet: 0.11 Normalized: 0.43 UNIX> echo "0.25 7 4 .25 5 .75 6 75 7 3000"  keno Bet: 0.25 Balls Picked: 7 Probability of catching 4 of 7: 0.052191  Expected return: 0.0130477 Probability of catching 5 of 7: 0.0086385  Expected return: 0.00647888 Probability of catching 6 of 7: 0.000732077  Expected return: 0.0549058 Probability of catching 7 of 7: 2.44026e05  Expected return: 0.0732077 Your return per bet: 0.10 Normalized: 0.41 UNIX> echo "0.25 8 5 1 6 20 7 300 8 10000"  keno Bet: 0.25 Balls Picked: 8 Probability of catching 5 of 8: 0.0183026  Expected return: 0.0183026 Probability of catching 6 of 8: 0.00236671  Expected return: 0.0473343 Probability of catching 7 of 8: 0.000160455  Expected return: 0.0481365 Probability of catching 8 of 8: 4.34566e06  Expected return: 0.0434566 Your return per bet: 0.09 Normalized: 0.37 UNIX> echo "0.25 9 5 1 6 10 7 35 8 800 9 13000"  keno Bet: 0.25 Balls Picked: 9 Probability of catching 5 of 9: 0.0326015  Expected return: 0.0326015 Probability of catching 6 of 9: 0.00571956  Expected return: 0.0571956 Probability of catching 7 of 9: 0.000591678  Expected return: 0.0207087 Probability of catching 8 of 9: 3.25925e05  Expected return: 0.026074 Probability of catching 9 of 9: 7.24277e07  Expected return: 0.0094156 Your return per bet: 0.10 Normalized: 0.42 UNIX> echo "0.25 10 5 .25 6 2.50 7 25 8 250 9 2500 10 25000"  keno Bet: 0.25 Balls Picked: 10 Probability of catching 5 of 10: 0.0514277  Expected return: 0.0128569 Probability of catching 6 of 10: 0.0114794  Expected return: 0.0286985 Probability of catching 7 of 10: 0.00161114  Expected return: 0.0402786 Probability of catching 8 of 10: 0.000135419  Expected return: 0.0338548 Probability of catching 9 of 10: 6.12065e06  Expected return: 0.0153016 Probability of catching 10 of 10: 1.12212e07  Expected return: 0.0028053 Your return per bet: 0.12 Normalized: 0.46 UNIX>Clearly, the "2bit" menu is not a sound investment strategy. I especially love the part that says "Don't know what number to play? We have Quick Pick!" How about you just remove money from my bank account and then I don't have to think at all!!!!!!
How about the other Keno games? "You asked for it  The Catch All 5 Spot":
UNIX> echo "1.50 5 5 1300"  keno Bet: 1.50 Balls Picked: 5 Probability of catching 5 of 5: 0.000644925  Expected return: 0.838402 Your return per bet: 0.66 Normalized: 0.44 UNIX>The "Brand New 8 spot:"
UNIX> echo ".40 8 5 2 6 20 7 200 8 20000"  keno Bet: 0.40 Balls Picked: 8 Probability of catching 5 of 8: 0.0183026  Expected return: 0.0366052 Probability of catching 6 of 8: 0.00236671  Expected return: 0.0473343 Probability of catching 7 of 8: 0.000160455  Expected return: 0.032091 Probability of catching 8 of 8: 4.34566e06  Expected return: 0.0869132 Your return per bet: 0.20 Normalized: 0.49 UNIX>And "100 Dimes"  note this one has a catch zero:
UNIX> echo ".10 7 0 .10 6 20 7 1200"  keno Bet: 0.10 Balls Picked: 7 Probability of catching 0 of 7: 0.121574  Expected return: 0.0121574 Probability of catching 6 of 7: 0.000732077  Expected return: 0.0146415 Probability of catching 7 of 7: 2.44026e05  Expected return: 0.0292831 Your return per bet: 0.04 Normalized: 0.44 UNIX>Perhaps I should have ridden the gondola........
Using your newfound understanding of Keno and your ability to program, write up a paragraph that shows Pippy how much she is losing on each of her fifteen potential Keno bets.
UNIX> echo "0.25 5 3 0.25 4 1 5 200"  keno1 Bet: 0.25 Balls Picked: 5 Catch 3/5  0.25 Catch 4/5  1.00 Catch 5/5  200.00 UNIX>Stage 2 calculates the probabilities.
UNIX> echo "0.25 5 3 0.25 4 1 5 200"  keno2 Bet: 0.25 Balls Picked: 5 Probability of catching 3 of 5: 0.0839351 Probability of catching 4 of 5: 0.0120923 Probability of catching 5 of 5: 0.000644925 UNIX>And then stage three finishes everything up.