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THE POWERBALL LOTTERY

lottery ticket

PROBABILITY
A probability is a measure of chance.
A probability is a number between 0 and 1 inclusive.
If it's more or less, you counted wrong!
The bigger the number, the better the chance.
The smaller the number, the less of a chance.
A probability is the quotient of two counts:
The number of ways something can happen that you want
divided by
The total number of ways something can happen
whether you like it or not.
We can concoct a probability formula like this;

P(x) = WAYS WANTED / WAYS TOTAL
P(x) = YOUR COUNT / TOTAL COUNT
P(x) = SOME WAYS / ALL WAYS,

where P(x) means the probability of an event x.
For example: what is the probability of drawing an Ace from a deck of cards.
There are 4 ways to draw an Ace from a deck of cards with 52 cards
4 ways you want and 52 ways total.
Therefore, P(Ace) = 4/52 = .077

There are thus 48 ways not to draw an Ace against 4 ways to draw one.
We call these the Odds against drawing an Ace.
We write Odds like this 48 to 4 or 48:4.

Since the number of some is always less than or equal to the number of all,
probability is always less than or equal to 1,
but probability cannot be negative either.
So probability can never be less than 0.
For more about probability,
read Chapter 8 in your text.
Read also the Wikipedia on Probability.

A PICK 5

POWERBALL PRIZES


IF YOU MATCH YOU WIN
5 of 5 + POWERBALL  JACKPOT
5 of 5  $200,000
4 of 5 + POWERBALL $10,000 
4 of 5 $100 
3 of 5 + POWERBALL  $100 
3 of 5  $7 
2 of 5 + POWERBALL  $7 
1 of 5 + POWERBALL  $4 
POWERBALL ONLY $3

PLAY ASSIGNMENT

Play the Tennessee Powerball Lottery 100 times in the mathlet ticket.
You may play the same set of numbers 100 times.
The numbers 1 to 55 are called the Pick numbers.
The numbers 1 to 42 are the Powerball numbers.
To win the Jackpot, you must guess the correct 5 numbers from 55,
plus pick the correct Powerball number.
Does it matter what order you pick the numbers?

Next play the Miniball Lottery ticket 100 times with less prizes.
Notice: there are less numbers in Miniball.
In which lottery will you win the most money?
Screen copy each ticket after you have played each 100 times
and email them to your instructor.
Then move on to fill in the tables below.

MINIBALL ASSIGNMENT

Before answering questions about the Tennessee Powerball Lottery, consider a simpler lottery first. Assume that a smaller lottery had only 6 numbers and you must pick 3 plus 1 of 10 possible Powerball numbers. The lottery ticket for Miniball would look something like this:

B PICK 3

MINIBALL PRIZES


IF YOU MATCH YOU WIN
3 of 3 + POWERBALL  $50 JACKPOT
3 of 3  $7
2 of 3 + POWERBALL  $5 
1 of 3 + POWERBALL  $3 
POWERBALL ONLY $1

MINIBALL LIST: PICK 3

In the text area below, list all the possible ways you can pick 3 numbers out of 6 numbers. The numbers in Miniball are 1 2 3 4 5 6. Keep these questions in mind when making your list: Is repetition allowed in Miniball? Can you pick the same number twice in Miniball? Does the order you pick the numbers matter in Miniball? How many ways should be on your list? Screencopy and email your list to your instructor.

MINIBALL TABLE 1: PICK 3

Below are two probability distribution tables for Miniball. For 5 points each, fill out each table. You will need to fill out the list above first.

Table 1 only considers the top 6 numbers, and how many ways you can choose 3 of them. Ignore the Powerball. Use the list you previously made to fill in each cell of the table. The x column lists the ways matches can vary. When picking 3 numbers, you could match 0 to 3 of them. The freqs column should contain the number of ways each possible match can be made. How frequently out of all the possible ways can 0 matches be made? Exactly 1 match? 2 matches? 3? The P(x) column should list the probability for each match. Enter probabilities as two significant digits. (Table 2 requires three significant digits for this column.) In the Fair $$$ column, enter the fair dollar amount of the bet. A fair bet is the dollar amount you should be paid to make the bet fair. Receiving $50 for the Jackpot is not a fair bet if you paid $1 to play. With a little experimenting, you should see the pattern quickly enough.

After filling out rows 0 to 3, sum up each column. If your entry is correct, it will turn blue when you press the ENTER key. If wrong, your entry will turn red. Clicking the mouse in a table cell, erases your previous entry. When completed, screen copy your tables and email them to your instructor.

MINIBALL TABLE 2: PICK 3 + POWERBALL

In Table 2, the randomness of x varies more. Table 2 includes all the possibilities of the Pick 3 numbers plus the ten possible Powerballs. The +B indicates number of matches plus the Powerball. The -B indicates number of matches minus the Powerball. The P(x) column requires 3 significant digits. After filling out all the rows, sum up each column. If your entry is correct, it will turn blue when you press the ENTER key. If wrong, your entry will turn red. Clicking the mouse in a table cell, erases your previous entry. When completed, screen copy your tables and email them to your instructor. Then move on to answer questions about Tennessee Powerball.

THE TENNESSEE POWERBALL

How many ways can you pick 5 numbers out of the 55 choices in one play?
Assume one play costs one dollar. How much will you need to spend to guarantee winning the jackpot?

Assume you play as many different possibilities as required to guarantee winning the jackpot. (A different possibility differs by AT LEAST one number from all the other possible sets of six numbers.) Then...
How many times should you expect to spend one dollar before you win the jackpot?
How many times should you expect to win the 50 million dollar jackpot?
What is your expected net winnings? (Net $Profit = $Won - $Lost)
Average your expected net winnings over the number of times you played. To the nearest cent, what is your average gain or loss?
How many different plays do you need to buy for a 50% chance of winning the jackpot?

Assume it takes one second to play the Tennessee Powerball Lottery mathlet when you click the QP button...
How many hours, days, months, years, centuries, or millennia will it take you to play for each dollar you spent in question 2? (Calculate the biggest time unit that yields the smallest positive number rounded off to the nearest INTEGER. Don't round off till the last step. Millennia are bigger time units than seconds.)
If the Jackpot were $10000, and the tickets cost nothing, would you play? (YES or NO)
If you play once for each ticket you bought in question 2, and you have not won yet, what is the probability you will win on the very next play? (Round off answer to 2 significant digits. Look up Significant Digits at Wikipedia.org. )

Enter your answers in the table below. Enter probabilities as decimal numbers. You may need to make lists to check your answers. Your answers may require negative signs. When done, screen copy the table, and email them to your instructor. Before you receive any points, you must play the lottery games and complete the tables above. And may all your bets be good ones!

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Last Modified 11/6/08 3:58 PM