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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
About 1/ 1,000,000
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
    someone will answer!
  • 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!
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