1. Machine Learning Group University College Dublin Statistics Overview Probability Conditional Probability Probability Density Functions

2. Probability 2 Probability: roots in gambling ­I’ll see you. What’ve you got? ­Five aces - and you? ­Two revolvers. ­All right, you win.

3. Probability 3 Easy Stuff Coin toss  Probability of heads is 1/2 Fair die  Probability of throwing any number is 1/6 Bag of marbles (2 red, 3 green, 4 blue)  Probability of drawing a red marble is 2/9  Probability of not drawing blue is:

4. Probability 4 Still Easy Stuff Joe and Fred play tennis each week  So far this year Joe has won 12 times and Fred 18 What is the chance of Joe winning tomorrow?  Joe also plays Mary and wins 1 of 4 games. What is the chance of Joe winning both games this week? What if Joe has good weeks and bad weeks?

5. Probability 5 Basic Rules These are examples of the addition and multiplication rules  The probability of the union of two independent events is the sum of their individual probabilities. we can also extend this to cover non-disjoint events  The probability of the intersection of two independent events is the product of their probabilities.

6. Probability 6 Some less intuitive stuff… How many cards do you need to be dealt before the odds of getting a pair are better than ½ ?

7. Probability 7 Simple Model: Cards replaced This model overestimates if cards are not replaced – chances are less

8. Probability 8 Correct Model

9. Probability 9 Birthday Example What are the chances in a group of n people of two having the same birthday?

10. Probability 10 Conditional Probability Draw 2 cards from a deck  What is the probability that the second is the 7  ? 1/52 – in the absence of other info  If the first card is not the 7  then the odds are 1/51, otherwise they are 0. What are the odds of getting 4 Aces? What are the odds of getting a Royal Flush? What are the odds of 4 Aces and a Royal Flush turning up together?

11. Probability 11 Back to Joe… Joe plays better on windy days He also plays better on wet days  What are his chances against Mary on windy wet days? What if wet days are inclined to be windy?

12. Probability 12 Conditional Probability Say I want to go kayaking and I want to ring the friends most likely to come along:  Data from the last 20 weekends: Jim 9 Síle 10 Brian 12 Brian is most likely to come out: 0.6 probability

13. Probability 13 Conditional Probability contd… LowMedHighTot Jim2439 Síle24410 Brian65112 Tot78520 However, if we have data on water levels and high water Is explected…

14. Probability 14 Conditional Probability contd… LowMedHighTot Jim2439 Síle24410 Brian65112 Tot78520 However, if we have data on water levels and high water is explected… reads as “probability of J given H”

15. Probability 15 Conditional Probability contd… Given two events J and H Then using the multiplication rule for independent events Probability of J given H is the same as proability of J iff J and H independent

16. Probability 16 Conditional Probability exercise LowMedHighTot Jim2439 Síle24410 Brian65112 Tot78520 Calculate P[J|H], P[H], P[J  H]

17. Probability 17 Blood Type Problem TA A O AB B O  TA A  TA B  TA AB  TA Blood type distribution for the US is A 41%, B9%, AB 4%, O 46%. In testing 88% with type A are correctly typed, 4% of Bs are typed as A, 10% of ABs are typed as A and 4% of Os are typed as A, What is the Probability that a guy typed as A is correctly typed. P[A] = 0.41 P[B]= 0.09 P[AB] = 0.04 P[O] = 0.46 P[TA|A] = 0.88 P[TA|B]= 0.04 P[TA|AB] = 0.1 P[TA|O] = 0.04

18. Probability 18 Bayes Theorem For the blood example, let or

19. Probability 19 Probability Density Functions For a discrete variable such as water level (Low, Med, High).

20. Probability 20 Continuous Density Functions In fact, water level is a continuous variable. Most common distribution is the Normal distribution. Where mis the mean and s is The standard deviation. A special case is the standard normal dist. where  is 0 and  is 1.

21. Probability 21 Summary Probability  The addition rule  The multiplication rule (independent events) Conditional probability  When events are not independent  Bayes Theorem Probability Density Functions