Randomness and Probability Randomness and Probability models Probability and Randomness Sample spaces Probability properties Assigning probabilities equally likely outcomes Assigning probabilities finite number of outcomes Probability rules mutually exclusive vs independence Probability 1 What is the probability that a flipped coin comes up heads 2 What are the odds that Microsoft stock price will go up tomorrow 3 What is the chance of rolling a 3 or 4 on a die Randomness and probability A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions The probability of any outcome of a random phenomenon can be defined as the proportion of times the outcome would occur in a very long series of repetitions Coin toss The result of any single coin toss is random But the result over many tosses is predictable as long as the trials are independent i e the outcome of a new coin flip is not influenced by the result of the previous flip The probability of heads is 0 5 the proportion of times you get heads in many repeated trials First series of tosses Second series The result of any single coin toss is random But the result over many tosses is predictable as long as the trials are independent i e the outcome of a new coin flip is not influenced by the result of the previous flip Probability models Probability models describe mathematically the outcome of random processes They consist of two parts 1 S Sample Space This is a set or list of all possible outcomes of a random process An event is a subset of the sample space 2 A probability for each possible event in the sample space S Example Probability Model for a Coin Toss S Head Tail Probability of heads 0 5 Probability of tails 0 5 Sample Space Every possible outcome AKA Universe or population S or A S outcomes Probability P A Size of the Event A Size of the Sample Space S General Purpose Definition Simple Case If outcomes are equally likely P A outcomes in A Total outcomes P Heads P Draw a King 4 52 1 13 Flip a Coin Three Times Outcomes HHH HHT HTH HTT THH THT TTH TTT 1 P HHH 1 8 0 125 2 P Two Heads 3 P At least 2 Heads 3 8 0 375 4 8 0 5 Outcomes Roll Two Dice 1 3 1 2 2 3 2 2 1 1 2 1 1 4 2 4 1 5 2 5 1 6 2 6 3 1 3 2 3 3 3 4 3 5 3 6 4 1 4 2 4 3 4 4 4 5 4 6 5 1 5 2 5 3 5 4 5 5 5 6 6 1 6 2 6 3 6 4 6 5 6 6 P Sum 2 1 36 0 0278 P Sum 9 4 36 1 9 0 111 P Sum 7 6 36 1 6 0 167 The gambling industry relies on probability distributions to calculate the odds of winning The rewards are then fixed precisely so that on average players lose and the house wins The industry is very tough on so called cheaters because their probability to win exceeds that of the house Remember that it is a business and therefore it has to be profitable Probability Properties Probabilities range from 0 no chance of the event to 1 the event has to happen 0 P A 1 P A 0 Because some outcome must occur A is impossible on every trial the sum of the probabilities for all possible outcomes the sample space must be exactly 1 P S 1 The Addition OR Rule Mutually Exclusive Events contain no common outcomes Intersection is empty They can t both happen For mutually exclusive events A B P A or B P A P B Mutually Exclusive Mutually Exclusive Roll a 2 and a 6 Draw a King and a Queen Flip two heads and 1 head Not Mutually Exclusive Roll a 2 and then a 6 Draw a King and a Spade Flip two heads and at least 1 head OR Rule Roll 2 dice P Sum is 7 or 9 1 6 1 9 5 18 Flip three coins P 1 H or 3 H 3 8 1 8 Draw a card P K or Q 1 13 1 13 2 13 P Diamond or Heart 1 2 P K or Diamond Simple Events Simple event contains one outcome Heads HHH 1 1 King of Spades Simple events are mutually exclusive Equally likely P Simple Event 1 total of outcomes Unequal Outcomes Assign a probability to each outcome All probabilities 0 P A sum P of each outcome in A All probabilities sum to 1 P S 1 All probabilities 1 M M candies If you draw an M M candy at random from a bag the candy will have one of six colors The probability of drawing each color depends on the proportions manufactured as described here Color Brown Red Yellow Green Orange Blue Probability 0 3 0 2 0 2 0 1 0 1 What is the probability that an M M chosen at random is blue S brown red yellow green orange blue P S P brown P red P yellow P green P orange P blue 1 P blue 1 P brown P red P yellow P green P orange 1 0 3 0 2 0 2 0 1 0 1 0 1 What is the probability that a random M M is either red yellow or orange P red or yellow or orange P red P yellow P orange 0 2 0 2 0 1 0 5 Complement Rule Complement of A All outcomes not in A Ac P Ac 1 P A P Drawing a card other than an Ace 1 1 13 12 13 General Addition OR Rule For any events A B P A or B P A P B P A and B P King 4 52 1 13 P Heart 13 52 P King and Heart P King of Hearts 1 52 P King or Heart P King P Heart P King and Heart 4 52 13 52 1 52 16 52 4 13 Subjective There is a 30 chance of rain of tomorrow What does that mean What is the Sample Space What are the outcomes Subjective Probability An opinion or judgment by a decision maker about the likelihood of an event based on their expertise Independence A and B independent Are not related Knowing A does not give information about B A does not affect B Two coin flips Roll a die twice Draw two cards Weather on two consecutive days Multiplication AND Rule P A and B P A P B If and only if A and B are independent Roll a die two times P roll a 3 and then a 5 1 6 1 6 1 36 P roll an even number twice 1 2 1 2 1 4 Independent repetition of game Choose a number between 1 and 10 A …
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