Probability 1. Joseph Spring School of Computer Science. SSP and Probability


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1 Probability 1 Joseph Spring School of Computer Science SSP and Probability
2 Areas for Discussion Experimental v Theoretical Probability Looking Back v Looking Forward Theoretical Probability Sample Space, Event, Outcome, Expectation & Variance Fundamental Probability Models Coins, Cards & Dice The Three Addition Laws, Tree Diagrams Independent & Conditional (Dependent) Events
3 Probability A Definition a measure of our faith in the likelihood of an event occurring Two types of probability Experimental to establish fairness / bias measures the past Theoretical as a predictor a predictor for the future
4 Experimental v Theoretical Probability Experimental Probability Looking back Method (e.g. a fair coin?) Collect Data use relative frequency for experimental probability p(head) = number of times head occurs total number of throws Example throw a coin 100 times and record cumulative frequency experimental frequency
5 Experimental v Theoretical Probability Number of throws (n) Result h t t t h Number of heads (f) (cum ulative) Experimental probability (f/n)
6 Experimental v Theoretical Probability Theoretical Probability Looking forward a predictor probability taken as: p( an event) = number of ways of gaining success total number of options available
7 Terms & Probability Sample Space the set of possible outcomes from a trial or experiment the set will cover all possible outcomes each possible outcome will correspond to only one member of the sample space Example Ω={ picture cards, red cards, black cards that are not picture cards } the above example would not be a sample space since if you selected the King of Hearts this would correspond to picture cards and also red cards
8 Terms & Probability Example Ω={ red cards, black cards } the above example would be a sample space since a card is either a black card or a red card but not both
9 Theoretical Probability Event an event is a possible outcome resulting from an experiment for example selecting a red card Outcome acceptable things that can happen in an experiment e.g. fair coin (head or tail only) no landing on the thin edge of the coin
10 Theoretical Probability Expectation ( or Expected Value) the theoretical analogue of mean average Expectation = n i= 1 px Question What is the expectation when throwing a fair die? Answer 3.5 (why?) Question How are expectation and experimental mean average related? Answer frequency/n becomes probability, so... i i
11 Theoretical Probability Variance n n 2 2 px i i px i i i= 1 i= 1 Variance = ( ) Question How does this relate to experimental variance? Answer frequency/n becomes probability, so...
12 Fundamental Probability Models Coins Cards Dice
13 The Three Addition Laws 1. PA ( B) = PA ( ) + PB ( ) PA ( B) 2. PA ( B) = PA ( ) + PB ( ) for mutually exclusive (M.E.) events _ 3. P(A) = 1 P( A)
14 Independent & Conditional Events Example Select a card from a well shuffled pack of playing cards. Read the card. Now replace it in the pack, shuffle well and select another. These two selections are independent If the card is not replaced then the two selections will be conditional events
15 Independent & Conditional Events Independent Events two events are said to be independent if the outcome of one event does not affect the probability for the outcome of the other event Conditional Events two events are said to be conditional if the outcome of one event does affect the probability for the outcome of the other event
16 Tree Diagrams Independent Events P(King) = 4/52 King P(K) = 4/52 P(not K) = 48/52 King P(not K) = 48/52 P(K) = 4/52 not a King P(not K) = 48/52 not a King King not a King
17 Tree Diagrams Conditional (Dependent) Events P(King) = 3/51 King P(K) = 4/52 P(not K) = 48/52 King P(not K) = 48/51 P(K) = 4/51 not a King P(not K) = 47/51 not a King King not a King
18 Summary Experimental v Theoretical Probability Looking Back v Looking Forward Theoretical Probability Sample Space, Event, Outcome, Expectation & Variance Fundamental Probability Models Coins, Cards & Dice The Three Addition Laws, Tree Diagrams Independent & Conditional (Dependent) Events
When combined events A and B are independent:
A Resource for reestanding Mathematics Qualifications A or B Mutually exclusive means that A and B cannot both happen at the same time. Venn Diagram showing mutually exclusive events: Aces The events
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