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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 Docsity.com
(We're repeating concepts from the last lecture) There is a close relationship between probability and counting Counting and probability 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 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 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 Docsity.com
Every run is different.
Empirical probabilities vary. What happens:
- 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/
