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## Gambles in Your Life

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### UO Xmas Lottery! Everyone has a ticket. We will draw from a box to choose the winner. ... UO Xmas Birthday Giveaway! First two birthdays to match win! We know ... – PowerPoint PPT presentation

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Title: Gambles in Your Life

1
Gambles in Your Life
• Andre Dabrowski
• Mathematics and Statistics

2
Pick the Prize!
3
One Chance in Three
ProbWinner (winning choices) / (all
possible choices) / (
) 1/3
4
Probevent
5
Gambles in your Life
• Pwinner
• (winning choices)/(all choices)
• 1/(all choices)

6
Lottery 216
• Everyone has a sample ticket.
• Every ticket has 3 numbers, each number chosen
from 1,2,3,4,5,6.E.G. 136 or 524 or 652, but
not 744.
• Is 222 more or less likely to win than 452?
• What is your chance of winning?

7
Is 222 more or less likely to win than 452?
• Put one marker in the box for each ticket.
• Mix them up.
• Draw one out.
• All tickets have the same chance at winning!
• So 222 has the same chance as 452 of winning.

8
What is your chance of winning?
• Pwinner1/(all possible choices)
• (all possible choices)
• (choices for first digit) X (choices for
second digit) X (choices for third digit)
• 6 X 6 X 6 216
• Pwinner1/216.

9
Lotto 6/49
• Pwin by matching all 6 numbers
• 1/(all possible choices)
• (all possible choices)
• 49 x 48 x 47 x 46 x 45 x 44 / 720
• 1 in 13,983,816 chances!

10
Which is more likely?
• Matching all 6 numbers in a 6/49 lottery
• Being struck by lightning sometime during the
year.

1/ 13,983,816
11
(No Transcript)
12
UO Xmas Lottery!
• Everyone has a ticket.
• We will draw from a box to choose the winner.
• Pwinning1/216.

13
Now that we know HOW to calculate probabilities,
we can look for interesting ones to compute.
14
The Birthday Problem
• There are 365 days in the year.
• The chance that any one person shares your
birthday is 1/365. Pretty small!
• What is the chance at least two people in this
room share birthdays?

15
Pno matching birthdays
• Pno match for 2 people
• 365 X 364 / 365 X 365 364/365.

16
• Pno match in 5 people

.97 approximately
17
• Pno match in 25 people

0.43 approximately
There is about a 57 chance a class of 25 will
have at least two sharing a birthday.
18
• Pbirthday match in k people

19
Gambles in your Life
• Small probabilities can become large if we do
many simultaneous experiments.
• Coincidences are not really coincidences in
large groups. Yell Hey Pete in a crowd and
• How reliable are complex systems? A system
can survive one component failing, but what is
the chance two fail at once?

20
UO Xmas Birthday Giveaway!
First two birthdays to match win!
21
• We know
• How to compute probabilities for simple games
• How do we compute probabilities for more
complicated problems?

22
Simple --8 Heads in a Row in 8 tosses
• Chance of 8 heads in a row
• ½ ? ½ ? ? ½ ? ½ ? ½
• 1/256
• Pretty small!

23
Harder -- 8 Heads in a Row somewhere in 100
tosses
• Toss a fair coin 100 times.
• What is the chance of at least 8 heads in a row
somewhere in the string of 100?
• HHTHTTHTTTHTHHTHTHTH TTTHTHTHTHHTHHHHHHHH
TTHHHTHTHTHHHTTHTHTH HHTHTHHTHTHTTTHTTTT
HT THTTHTHTHHHTHTHTHTHTT

24
Monte Carlo Methods
• Toss a coin 100 times
• Find the longest string of Hs
• Repeat this 100,000 times --- 10,000,000
tosses!
• Pat least 8 Hs in a row is approximately
• ( at least 8 Hs in a row)/100,000

25
Monte Carlo Methods
• Toss a coin 100 times using a computer
• Find the longest string of HsRepeat this
100,000 times --- 10,000,000 tosses! using a
computer
• Pat least 8 Hs in a row is approximately(at
least 8 Hs in a row)/100,000

26
Monte Carlo Methods
• libname here 'h/XmasLecture'
• libname there 'c/tmp'
• macro dupit
• do ii1 to 100
• x_ii(ranuni(0)lt.5)
• end
• mend
• data there.runs
• do i1 to 100000 by 1
• output
• end
• data there.runs set there.runs
• dupit
• run
• macro runs
• do ii2 to 100
• let iiieval(ii-1)
• a0run_iii
• data there.runs set there.runs
• run_10(x_11)
• runmax0
• runs
• data here.runs set there.runs
• keep runmax
• run
• data there.runs run
• proc gchart datahere.runs
• axis1 value(height10)
• vbar runmax / midpoints 1 to 15 by 1
typepercent caxisaxis1
• run
• proc freq datahere.runs
• table runmax / nofreq nocumulative
• run
• quit

27
100,000 Simulations
28
The FREQ Procedure runmax Frequency
Percent ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 2
23 0.02 3 2709 2.71 4
16421 16.42 5 26184 26.18 6
23039 23.04 7 14645 14.65 gt7
16979 16.98 All 100000 100.00

Prun of 8 H or more .17 approx., 1/6 gtgt
1/256.
The chance of 4 or more heads in a row is about
97. We can use this to pick out which sequences
on the sheet are unlikely to really have been
generated at random.
29
Gambles in your Life
• good days and bad days.
• Long lineups for no reason.
• Design of bridges, power plants.
• Weather prediction.
• Biological evolution.

30
Merry Christmas and a Happy New Year
Thanks for coming!