**Class 18: Box Models**

**Group work:**

Translate
each of the following situations into a statement about a box model by filling
in the blanks. For each situation,
think about a) what numbers should go on the tickets, b) how many tickets of
each type should go into the box, and c) how many times you draw a ticket.

This
is like the sum of _______________ draws from the box containing the numbered
tickets ________________________________________.

1)
Flip a coin 100 times, and count the number of
heads.

2)
Roll 5 dice and count the total number of spots.

3)
Roll 5 dice and count the number of 6Õs.

4)
Flip a coin 10 times. Win $5 for each head that comes up and lose $5 for each tail
that comes up.

5)
Play roulette. In roulette, there are 38 squares. 18 squares are red, 18 are black, and 2 are green. Play $1 to play each time. Each time, bet on the number 13. If you win, you get your dollar back
plus $35. If you lose, you just
lose your dollar. Play 20 times.

6)
Play roulette again 100 times, this time betting
$1 each time on reds. If you win,
you get your dollar back plus a dollar.
If you lose, you lose your dollar.

7)
Challenge:
Play this game. Pay $1 to
play. Toss a coin twice. If two heads come up, you win right
then. If you get anything else
besides two heads, you get a second chance: toss the coin twice more, and if
you get two heads you win. If you
fail to get two heads on the second chance, you lose. Pay $1 to play.
If you win, you get your dollar back, plus another dollar.

**Homework Assignment
for Thurs, Oct 27: **On blackboard (Ch 16 and part of Ch 17)

**Announcements: **WeÕll have a Quiz on Thursday, Oct. 29, covering
probability and the binomial formula.

**Journal****:**

The California lottery includes several different
games. LetÕs focus on the Fantasy
5. The web site http://www.calottery.com/Games/FantasyFive/HowtoPlay/
describes how to play the game. You
get to pick any 5 numbers from 1 to 39.
You **cannot repeat numbers** in
the Fantasy 5, so the sequence of five numbers 3, 13, 33, 1, 7 is allowed, but
3, 4, 6, 4, 22 is not allowed because the number 4 is picked twice. You win the grand prize if you match the 5 numbers in the
drawing, in any order. So if you
pick 3,13, 33, 1, 7 and the draw is 1, 7, 33, 3, 13, then you win the grand
prize.

1) How many ways are there to pick 5 different numbers from 1 to 39 when the order of the numbers matters? (Hint: there are 39 ways to pick the first number, and 38 ways to pick a second number different from the first, and 37 ways to pick a third number different from the first two, etc.).

2) But in the Fantasy 5, you win the grand prize if you match all 5 numbers, even if they are in a different order. How many ways are there to pick 5 different numbers if the order doesnÕt matter? (Hint: take your answer to #1 and divide by 5!, since 5! Is the number of ways to arrange the 5 numbers in different orders.)

3) The web page http://www.calottery.com/Games/FantasyFive/HowtoPlay/ says that the odds of matching all 5 of 5 winning numbers is 1 in 575,757. Where did they get this number from?

4) Check the website www.calottery.com for the jackpot estimate for Fantasy 5 today and write it down. For example, the jackpot on Oct. 27 is estimated at $51,000.

5) Suppose you go out today buy 575, 757 tickets, one for each of the 575,757 combinations of numbers. Each ticket costs $1. How much money would you need to spend and how much money would you expect to win? Would you be guaranteed to win? (To simplify the math, pretend that you win the entire jackpot if you match all 5 numbers in any order, and win nothing if you only match 4 or 3 or 2 numbers.)

6) Suppose, instead, that you buy one ticket a day for 575,757 days. (To simplify the math, pretend that the jackpot is always the same size as todayÕs jackpot, and that it always pays the full jackpot if you match all 5 numbers, and nothing if you only match 4 or 3 or 2 numbers.) How much money would you expect to spend on tickets and about how much money would you expect to win? Would you be guaranteed to win? How old would you be at the end of these 575,757 days?

7) In fact, how much money you get if you win Fantasy 5 depends on how many people buy tickets that day, and the prize is split among all the winners. Why do you think the lottery is run this way, instead of offering the same prize of, say, $100,000 to each winner each time?

8) Do you have any lottery stories or comments youÕd like to share?