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0 means the event will never occur. 1 means the event always occurs. 0.5 means the event occurs roughly 1/2 of the time. The probability of a event is a number between 0 and 1 that determines "how likely" the event is to occur. 3 to 1 against means the probability of the event is 1/4 and the probability of the event not happening is 3/ 10 to 1 for means the probability of the event happening is 10 out of 11 times (10/11) while the probability of the event not happening is 1/11. You might have encountered probability stated as odds: Probability Friday, October 22, 2010 9:18 PM Docsity.com
(We're repeating concepts from the last lecture) There is a close relationship between probability and counting Counting and probability Wednesday, October 27, 2010 3:13 PM Docsity.com
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E.g., height and weight of people. E.g., flips of a coin. E.g., 5-card poker hands. An experiment is a situation in which you measure something. E.g., one person's height and weight. E.g., flipping a coin once. E.g., the contents of one poker hand. A trial of an experiment measures one situation E.g., heights and weights of all people. E.g., flipping a coin an unlimited number of times. E.g., all possible poker hands. A population is a set of possible outcomes for the experiment: E.g., heights and weights of 10 known people. E.g., flipping a coin 10 times. E.g., 10 poker hands. A sample is some subset of a population describe populations through samples. describe experiments through trials. Our goals Experiments, populations, and samples Monday, October 25, 2010 11:50 AM Docsity.com
We might roll a die, e.g., 100 times. What can we predict about the rolls? number of ways it can roll "six" = ---------------------------------------------------- total number of outcomes of one roll coming up with a "six" is 1/6 =. Theoretical probability 100 rolls might be: 2,2,4,6,4,5, 1 ,5,5,2,5, 1 ,5,4,6,5,2,4, 1 ,4, 5,2,5,2,4,5,4,3,2,2,5,3, 1 ,4,6,6, 1 ,3, 1 ,4, 3, 1 ,2, 1 ,3,4,2, 1 ,3,4,3,4,6,5,2,5, 1 ,6,3, 1 , 2, 1 ,2,5,4,2, 1 ,5,4,6,3,6,5,5,2,2,2,6,3,2, 5,6,3,2,2,4, 1 ,6, 1 ,2,4,2,4, 1 ,3,2,6, 1 ,6, In this there are 17 ones, so the empirical probability of rolling a one is 17/100 =. Empirical probability: Note: empirical probability almost never equals theoretical probability, unless the sample is large. Example: rolling a die Friday, October 22, 2010 9:23 PM Docsity.com
rand(1) generates a number between 0 and 1. rand(10) generates a float number between 0 and 10 (not including 10.0) int(rand(10)) generates an integer between 0 and 9. In Perl, the rand function generates a random number. int(rand(6))+1 is a random number between 1 and 6. so Simulating dice Monday, October 25, 2010 12:16 PM Docsity.com
Every run is different.
Empirical probabilities vary. 12:39 PM
Example: throwing two dice:
. "$howmany/36 = $prob\n";
make 100 rolls of two dice.
for ($i=0; $i<100; $i++) { $roll = int(rand(6))+1 + int(rand(6))+1; $stuff{$roll}++; # tabulate what happened }
show statistics
foreach my $s (2..12) { print "there are $stuff{$s} rolls of $s out of 100 \n"; $prob = $stuff{$s}/100; print " empirical probability of $s is $prob\n";
theoretical probability
$howmany=$s-1 if $s<=7; $howmany=13-$s if $s>7; $prob = $howmany/ 36; $prob = int($prob*100+0.5)/100; print " theoretical probability of $s is " } Pasted from Two dice Monday, October 25, 2010 12:41 PM Docsity.com
- there are 17 rolls of 1 out of
- empirical probability of 1 is 0.
- theoretical probability of 1 is 1/
- there are 24 rolls of 2 out of
- empirical probability of 2 is 0.
- theoretical probability of 2 is 1/
- there are 12 rolls of 3 out of
- empirical probability of 3 is 0.
- theoretical probability of 3 is 1/
- there are 17 rolls of 4 out of
- empirical probability of 4 is 0.
- theoretical probability of 4 is 1/
- there are 17 rolls of 5 out of
- empirical probability of 5 is 0.
- theoretical probability of 5 is 1/
- there are 13 rolls of 6 out of
- empirical probability of 6 is 0.
- theoretical probability of 6 is 1/
- Monday, October 25, What happens:
- there are 0 rolls of 2 out of
- empirical probability of 2 is
- theoretical probability of 2 is 1/36 = 0.
- there are 1 rolls of 3 out of
- empirical probability of 3 is 0.
- theoretical probability of 3 is 2/36 = 0.
- there are 12 rolls of 4 out of
- empirical probability of 4 is 0.
- theoretical probability of 4 is 3/36 = 0.
- there are 10 rolls of 5 out of
- empirical probability of 5 is 0.
- theoretical probability of 5 is 4/36 = 0.
- there are 12 rolls of 6 out of
- empirical probability of 6 is 0.
- theoretical probability of 6 is 5/36 = 0.
- there are 17 rolls of 7 out of
- empirical probability of 7 is 0.
- theoretical probability of 7 is 6/36 = 0.
- there are 17 rolls of 8 out of
- empirical probability of 8 is 0.
- theoretical probability of 8 is 5/36 = 0.
- there are 11 rolls of 9 out of
- empirical probability of 9 is 0.
- theoretical probability of 9 is 4/36 = 0.
- there are 9 rolls of 10 out of
- empirical probability of 10 is 0.
- Tuesday, October 26, What this does:
- empirical probability of 10 is 0.
- theoretical probability of 10 is 3/36 = 0.
- there are 9 rolls of 11 out of
- empirical probability of 11 is 0.
- theoretical probability of 11 is 2/36 = 0.
- there are 2 rolls of 12 out of
- empirical probability of 12 is 0.
- theoretical probability of 12 is 1/36 = 0.
roll 2 in 1 way (1+1) roll 3 in 2 ways (1+2, 2+1) roll 4 in 3 ways (1+3, 2+2, 3+1) roll 5 in 4 ways (1+4, 2+3, 3+2, 4+1) roll 6 in 5 ways (1+5, 2+4, 3+3, 4+2, 5+1) roll 7 in 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) roll 8 in 5 ways (2+6, 3+5, 4+4, 5+3, 6+2) roll 9 in 4 ways (3+6, 4+5, 5+4, 6+3) roll 10 in 3 ways (4+6, 5+5, 6+4) roll 11 in 2 ways (5+6, 6+5) roll 12 in 1 ways (6+6) one can So the theoretical probability of rolling a 7 is 6/36 = 1/6. 7 or 11 has a probability of 8/36 ("craps") Theoretical analysis of two dice Tuesday, October 26, 2010 3:37 PM Docsity.com
Repeat throws of two dice 100, 1000, 10000, 100000 times. What happens? We'll demonstrate the law of large numbers sunfire13{couch}113: perl large.perl after 100 trials: roll 2 3 4 5 6 7 8 9 10 11 12 got 0 4 10 9 15 17 17 14 7 2 5 expect 3 6 8 11 14 17 14 11 8 6 3 after 1000 trials: roll 2 3 4 5 6 7 8 9 10 11 12 got 19 63 92 109 130 170 140 104 85 51 37 expect 28 56 83 111 139 167 139 111 83 56 28 after 10000 trials: roll 2 3 4 5 6 7 8 9 10 11 12 got 273 574 838 1106 1359 1688 1430 1071 838 527 296 expect 278 556 833 1111 1389 1667 1389 1111 833 556 278 after 100000 trials: roll 2 3 4 5 6 7 8 9 10 11 12 got 2735 5565 8326 11212 13915 16553 13926 11009 8430 5552 2777 expect 2778 5556 8333 11111 13889 16667 13889 11111 8333 5556 2778 See http://www.cs.tufts.edu/comp/14/examples/Probability/large.perl As the number of rolls increases, the number of times each number appears approaches what theory suggests. Demonstrating the law of large numbers Monday, October 25, 2010 12:44 PM Docsity.com
If p(E) is the probability of E, and we "try" E n times, then we "expect" that E will be true np(E) times. Reason: p(E) = #successes/#trials = #successes/n, so np(E)=#successes. What we expect Tuesday, October 26, 2010 3:44 PM Docsity.com
Example: We draw one marble from an urn containing 3 red marbles, 1 blue, and 1 green. What is the probability that it is red? Answer: All outcomes S = {R 1 ,R 2 ,R 3 ,G,B} Desirable outcomes E = {R 1 ,R 2 ,R 3 } p(E)=|E|/|S| = 3/5. Example Tuesday, October 26, 2010 3:50 PM Docsity.com
Example: we draw three balls from the same urn. What is the probability that all are red? Answer: There are C(5,3) possible drawings, and C(3,3)= drawing that we desire, so the probability is |E|/|S|= 1/C(5,3)=1/10. Example Tuesday, October 26, 2010 3:52 PM Docsity.com
