USING LOTTERIES IN TEACHING A CHANCE COURSE.pdf

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USING LOTTERIES IN TEACHING A CHANCE COURSE
Written by the Chance Team for the Chance Teachers Guide
revised August 1, 1998
Probability is used in the Chance course in two ways. First, it is used to
help students understand issues in the news that rely on probability
concepts. These include: chances of winning at the lottery, streaks in
sports, random walk and the stock market, coincidences, evaluating extra
sensory perception claims, etc. Second, a knowledge of elementary
probability models, such as coin tossing, are necessary to understand
statistical concepts like margin of error for a poll and testing a
hypothesis.
Our goal here is to show how one can use current issues in the news and
various activities to make students appreciate the role that probability
plays in everyday news stories and to help them understand statistical
concepts.
We begin by illustrating this in terms of the many interesting probability
and statistics problems involved in lotteries. Lotteries are discussed
frequently in the news, and they have a huge impact directly and
indirectly on our lives. They are the most popular form of gambling and
an increasingly important way that states obtain revenue. In a Chance
course, we do not give the systematic account presented here, but rather
discuss a number of the points made in this presentation as they come
up in the news.
THE POWERBALL LOTTERY
We will discuss lotteries in terms of the Powerball Lottery. The
Powerball Lottery is a multi-state lottery, a format which is gaining
popularity because of the potential for large prizes. It is currently
available in 20 states and Washington D.C. It is run by the Multi-State
Lottery Association, and we shall use information from their web
homepage, http://www.musl.com. We found their "Frequently Asked
Questions," (hereafter abbreviated FAQ) to be particularly useful. These
are compiled by Charles Strutt, the executive director of the Association.
A Powerball lottery ticket costs $1. For each ticket you are asked mark
your choice of numbers in two boxes displayed as follows:
2
Pick 5
cash
annuity
01
10
18
26
34
42
Pick 1
01
08
15
22
29
36
02
09
16
23
30
37
03
10
17
24
31
38
04
11
18
25
32
39
05
12
19
26
33
40
06
13
20
27
34
41
02
11
19
27
35
43
03
12
20
28
36
44
04
13
21
29
37
45
05
14
22
30
38
46
06
15
23
31
39
47
07
16
24
32
40
48
08
17
25
33
41
49
EP__
09
EP__
07
14
21
28
35
42
Table 1: Picking your numbers.
You are asked to select five numbers from the top box and one from the
bottom box. The latter number is called the "Powerball". If you check
EP (Easy Pick) at the top of either box, the computer will make the
selections for you. You also must select "cash" or "annuity" to
determine how the jackpot will be paid should you win. In what
follows, we will refer to a particular selection of five plus one numbers
as a "pick."
Every Wednesday and Saturday night at 10:59 p.m. Eastern Time, lottery
officials draw five white balls out of a drum with 49 balls and one red
ball from a drum with 42 red balls. Players win prizes when the
numbers on their ticket match some or all of the numbers drawn (the
order in which the numbers are drawn does not matter). There are 9
ways to win. Here are the possible prizes as presented on the back of the
Powerball ticket:
3
You Match
5 white balls and the red ball
5 white balls but not the red ball
4 white balls and the red ball
4 white balls but not the red ball
3 white balls and the red ball
3 white balls but not the red ball
2 white balls and the red ball
1 white ball and the red ball
0 white balls and the red ball
You win
JACKPOT
*
$100,000
$5,000
$100
$100
$7
$7
$4
$3
Odds
1 in 80,089,128
1 in 1,953,393
1 in 364,041
1 in 8879
1 in 8466
1 in 206
1 in 605
1 in 118
1 in 74
Table 2: The chance of winning.
CALCULATING THE ODDS
The first question we ask is: how are these odds determined? This is a
counting problem that requires that you understand one simple
counting rule: if you can do one task in
n
ways and, for each of these,
another task in
m
ways, the number of ways the two tasks can be done
is
n
×
m
. A simple tree diagram makes this principle very clear.
When you watch the numbers being drawn on television, you see that,
as the five winning white balls come out of the drum, they are lined up
in a row. The first ball could be any one of 49. For each of these
possibilities the next ball could be any of 48, etc. Hence the number of
possibilities for the way the five white balls can come out in the order
drawn is 49
×
48
×
47
×
46
×
45 = 228,826,080.
But to win a prize, the order of these 5 white balls does not count. Thus,
for a particular set of 5 balls all possible orders are considered the same.
Again by our counting principle, there are 5
×
4
×
3
×
2
×
1 = 120 possible
orders. Thus, the number of possible sets of 5 white balls not counting
order is 228,826,080/120 = 1,906,884. This is the familiar problem of
choosing a set of 5 objects out of 49, and we denote this by C(49,5). Such
numbers are called binomial coefficients. We can express our result as:
49!
49
×
48
×
47
×
46
×
45
C(49,5) = 5! 44! =
5
×
4
×
3
×
2
×
1
Now for each pick of five white numbers there are 42 possibilities for
the red Powerball, so the total number of ways the winning six numbers
*
The official Lottery explanation for the Jackpot is: "Select the cash option and receive
the full cash amount in the prize pool. Select the annuity option and we will invest the
money and pay the annuity amount to you over 25 annual payments." The cash payment is
typically 50-60% of the total dollar amount paid over 25 years.
4
can be chosen is 42
×
C(49,5) = 80,089,128. We will need this number
often and denote it by
b
(for big).
The lottery officials go to great pains to make sure that all
b
possibilities
are equally likely. So, a player has one chance in 80,089,128 of winning
the jackpot. Of course, the player may have to share this prize.
We note that the last column in Table 2 is labeled "odds" when it more
properly describes the "probability of winning". Because the probabilities
are small, there is not much difference between odds and probabilities.
However, this is a good excuse to get the difference between the two
concepts straightened out. The media prefers to use odds, and textbooks
prefer to use probability or chance. Here the chance of winning the
jackpot is 1 in 80,089,128, whereas the odds are 1 to 80,089,127 in favor (or
80,089,127 to 1 against).
To win the $100,000 second prize, the player must get the 5 white
numbers correct but miss the Powerball number. How many ways can
this be accomplished? There is only one way to get the set of five white
numbers, but the player's Powerball pick can be any of the 41 numbers
different from the red number that was drawn. Thus, the chance of
winning second prize is 41 in 80,089,128; rounded to the nearest integer
this is 1 in 1,953,393.
This is a good time to introduce the concept of independence. You could
find the probability of winning the second prize by pointing out the
probability that you get the 5 white numbers correct is 1/C(49,5). The
chance of not getting the red ball correct is 41/42. Since these events are
independent, the chance that they both happen is the product of their
individual probabilities.
We can also point out that the lottery numbers you pick are independent
of those drawn to determine the winning numbers. On the other hand,
your picks and those of other buyers cannot be assumed to be
independent.
Discussion Question:
Why not?
Prior to November 2, 1997, the Powerball game was conducted by
drawing 5 white balls from a drum of 45 and one red powerball from a
second drum of 45. The prize for getting the red ball correct was $1, and
the ticket listed the chances as 1 in 84. This often seemed wrong to
players who have had elementary probability as the following exchange
from the Powerball FAQ
*
illustrates:
*
From the Multi-State Lottery Association web site at http://www.musl.com/
5
COULD YOUR ODDS BE WRONG?
I have a simple question. You list the odds of
matching only the powerball as one in 84 on the
powerball "ways to win" page. From my
understanding of statistics (I could be wrong, but I
got an A), the odds of selecting one number out of a
group is simply one over the number of choices.
Since there are not 84 choices for the powerball, may
I assume the balls are somehow "fixed" so that some
are more common than others? Otherwise, the listed
odds are somehow defying the laws of statistics. I am
really very eager to hear your explanation, so please
return my message. Thank you.
Susan G., via Internet.
This is one of most common questions we get about
the statistics of the game. If you could play only the
red Powerball, then your odds of matching it would
indeed be 1 in 45. But to win the $1 prize for
matching the red Powerball alone, you must do just
that; match the red Powerball ALONE. When you bet
a dollar and play the game, you might match one
white ball and the red Powerball. You might match
three white balls and the red Powerball. To determine
the probability of matching the red Powerball alone,
you have to factor in the chances of matching one or
more of the white balls too.
C.S.
To win this last prize you must choose your six numbers so that only the
Powerball number is correct. In the older version of the Powerball
lottery this would be done as follows: there are 45
×
C(45,5) = 54,979,155
ways to choose your six numbers. But here your first 5 numbers must
come from the 40 numbers
not
drawn by the lottery. This can happen in
C(40,5) = 658,008 ways. Now there is only one way to match the
Powerball number, so overall you have 658,008 chances out of 54,979,155
to win this prize. This reduces to 1 chance in 83.55, or about 1 chance in
4, in agreement with the official lottery pronouncement.
The same kind of reasoning of course carries over to the present version
of the game. To find the chance of winning any one of the prizes we
need only count the number of ways to win the prize and divide this by
the total number of possible picks
b.
Let
n
(i) be the number of ways to
win the ith prize. Then the values of
n
(i) are shown in Table 3 below.

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