gambles in your life
DESCRIPTION
Gambles in Your Life. Andre Dabrowski Mathematics and Statistics. Pick the Prize!. One Chance in Three. Prob[Winner] =#(winning choices) / #(all possible choices) = / #( ) = 1 / 3. Prob[event]. Gambles in your Life. P[winner] - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/1.jpg)
Gambles in Your LifeGambles in Your Life
Andre Dabrowski
Mathematics and Statistics
![Page 2: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/2.jpg)
Pick the Prize!
![Page 3: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/3.jpg)
One Chance in Three
Prob[Winner]
=#(winning choices) / #(all possible choices)
= / #( )
= 1/3
![Page 4: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/4.jpg)
Prob[event]Prob[event]
![Page 5: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/5.jpg)
Gambles in your Life• P[winner]
• =#(winning choices)/#(all choices)
• =1/#(all choices)
![Page 6: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/6.jpg)
“Lottery 216”1. Everyone has a sample ticket.
2. 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.
3. Is 222 more or less likely to win than 452?
4. What is your chance of winning?
![Page 7: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/7.jpg)
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.
![Page 8: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/8.jpg)
What is your chance of winning?
• P[winner]=1/#(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• P[winner]=1/216.
![Page 9: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/9.jpg)
Lotto 6/49
• P[win 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!
![Page 10: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/10.jpg)
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
![Page 11: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/11.jpg)
![Page 12: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/12.jpg)
UO Xmas Lottery!1. Everyone has a ticket.
2. We will draw from a box to choose the winner.
3. P[winning]=1/216.
![Page 13: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/13.jpg)
Now that we know HOW to calculate
probabilities, we can look for interesting ones to compute.
![Page 14: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/14.jpg)
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?
![Page 15: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/15.jpg)
P[no matching birthdays]
• P[no match for 2 people]
• =
• = 365 X 364 / 365 X 365 = 364/365.
#(ways of choosing 2 without matching)
#(ways of choosing 2 birthdays)
![Page 16: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/16.jpg)
• P[no match in 5 people]=
365 X 364 X 363 X 362 X 361
365 X 365 X 365 X 365 X 365
= .97 approximately
=
#(ways of choosing 5 without matching)
#(ways of choosing 5 birthdays)=
![Page 17: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/17.jpg)
• P[no match in 25 people]=
#(ways of choosing 25 without matching)
#(ways of choosing 25 birthdays)=
365 X 364 X … X 342 X 341
365 X 365 X … X 365 X 365=
= 0.43 approximately
There is about a 57% chance a class of 25 will have at least two sharing a birthday.
![Page 18: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/18.jpg)
• P[birthday match in k people]
![Page 19: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/19.jpg)
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?
![Page 20: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/20.jpg)
UO Xmas Birthday Giveaway!
First two birthdays to match win!
![Page 21: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/21.jpg)
We know
• How to compute probabilities for simple games
•How do we compute probabilities for more complicated problems?
![Page 22: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/22.jpg)
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!
![Page 23: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/23.jpg)
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 HHTHTHHTHTHTTTHTTTTHT THTTHTHTHHHTHTHTHTHTT
![Page 24: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/24.jpg)
Monte Carlo Methods• “Toss” a coin 100 times• Find the longest string of H’s• Repeat this 100,000 times
--- 10,000,000 tosses!• P[at least 8 H’s in a row] is
approximately • #( at least 8 H’s in a row)/100,000
![Page 25: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/25.jpg)
Monte Carlo Methods• “Toss” a coin 100 times
using a computer• Find the longest string of H’s
Repeat this 100,000 times --- 10,000,000 tosses!using a computer
• P[at least 8 H’s in a row] is approximately#(at least 8 H’s in a row)/100,000
![Page 26: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/26.jpg)
Monte Carlo Methods• libname here 'h:/XmasLecture';• libname there 'c:/tmp';• %macro dupit;• %do ii=1 %to 100;• x_&ii=(ranuni(0)<.5);• %end;• %mend;
• data there.runs;• do i=1 to 100000 by 1;• output;• end;• data there.runs; set there.runs;• %dupit;• run;• %macro runs;• %do ii=2 %to 100;• %let iii=%eval(&ii-1);• a=0+run_&iii;• b=0+x_ⅈ• run_&ii=a*(a>0)*(b=1)+(b=1);• runmax=max(runmax,run_&ii);• %end;• %mend;
•data there.runs; set there.runs;•run_1=0+(x_1=1);•runmax=0;•%runs;
•data here.runs; set there.runs;•keep runmax;•run;
•data there.runs; run;•proc gchart data=here.runs;•axis1 value=(height=10);•vbar runmax / midpoints = 1 to 15 by 1 type=percent caxis=axis1;•run;
•proc freq data=here.runs;•table runmax / nofreq nocumulative;•run;•quit;
![Page 27: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/27.jpg)
100,000 Simulations
![Page 28: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/28.jpg)
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
>7 16979 16.98
All 100000 100.00
P[run of 8 H or more]
= .17 approx.,
= 1/6 >> 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.
![Page 29: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/29.jpg)
Gambles in your Life• “good” days and “bad” days.
• Long lineups for no reason.
• Design of bridges, power plants.
• Weather prediction.
• Biological evolution.
![Page 30: Gambles in Your Life](https://reader036.vdocuments.us/reader036/viewer/2022062321/56812ac5550346895d8e9a28/html5/thumbnails/30.jpg)
Thanks for coming!