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Introduction to probability and Statistics

XOLISWA MASHIYANE
XOLISWA MASHIYANE

Probability is defined as the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible."the area under the curve represents probability".The slideshow is a very basic Introduction to probability,counting and sets.

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Welcome to 18.05 Introduction to Probability and Statistics Spring 2014 http://xkcd.com/904/ January 1, 2017 1 / 24
R Free open source package. Very easy to use and install. Instructions and a link for this are on MITx/18.05r. January 1, 2017 2 / 24
Platonic Dice January 1, 2017 3 / 24
Probability vs. Statistics Di鍖erent subjects: both about random processes Probability Logically self-contained A few rules for computing probabilities One correct answer Statistics Messier and more of an art Get experimental data and try to draw probabilistic conclusions No single correct answer January 1, 2017 4 / 24
Counting: Motivating Examples What is the probability of getting exactly 1 heads in 3 tosses of a fair coin? January 1, 2017 5 / 24
Poker Hands Deck of 52 cards 13 ranks: 2, 3, . . . , 9, 10, J, Q, K, A 4 suits: , , , , Poker hands Consists of 5 cards A one-pair hand consists of two cards having one rank and the remaining three cards having three other rank Example: {2, 2, 5, 8, K} The probability of a one-pair hand is: (1) less than 5% (2) between 5% and 10% (3) between 10% and 20% (4) between 20% and 40% (5) greater than 40% January 1, 2017 6 / 24
Sets in Words Old New England rule: dont eat clams (or any shell鍖sh) in months without an r in their name. S = all months L = the month has 31 days R = the month has an r in its name S = {Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec} L = {Jan, Mar, May, Jul, Aug, Oct, Dec} R = {Jan, Feb, Mar, Apr, Sep, Oct, Nov, Dec} L R = {Jan, Mar, Oct, Dec} = months with 31 days and an r January 1, 2017 7 / 24
Visualize Set Operations with Venn Diagrams S L R L R L R Lc L R January 1, 2017 8 / 24
Product of Sets S T = {(s, t)} January 1, 2017 9 / 24
Inclusion-Exclusion Principle S AB A B January 1, 2017 10 / 24
Board Question A band consists of singers and guitar players. 7 people sing 4 play guitar 2 do both How many people are in the band? January 1, 2017 11 / 24
Rule of Product 3 shirts, 4 pants = 12 out鍖ts (More powerful than it seems.) January 1, 2017 12 / 24
Concept Question: DNA DNA is made of sequences of nucleotides: A, C, G, T. How many DNA sequences of length 3 are there? (i) 12 (ii) 24 (iii) 64 (iv) 81 How many DNA sequences of length 3 are there with no repeats? (i) 12 (ii) 24 (iii) 64 (iv) 81 January 1, 2017 13 / 24
Board Question 1 There are 5 Competitors in 100m 鍖nal. How many ways can gold, silver, and bronze be awarded? January 1, 2017 14 / 24
Board Question 2 I wont wear green and red together; I think black or denim goes with anything; Here is my wardrobe. Shirts: 3B, 3R, 2G; sweaters 1B, 2R, 1G; pants 2D,2B. How many di鍖erent out鍖ts can I wear? January 1, 2017 15 / 24
Solution answer: Suppose we choose shirts 鍖rst. Depending on whether we choose red compatible or green compatible shirts there are di鍖erent numbers of sweaters we can choose next. So we split the problem up before using the rule of product. A multiplication tree is an easy way to present the answer. R B G R,B R,B,G B,G B, D B, D B, D 3 3 2 3 4 2 4 4 4 Shirts Sweaters Pants Multiplying down the paths of the tree: Number of out鍖ts = (3 3 4) + (3 4 4) + (2 2 4) = 100 January 1, 2017 16 / 24
Permutations Lining things up. How many ways can you do it? abc and cab are di鍖erent permutations of {a, b, c} January 1, 2017 17 / 24
Permutations of k from a set of n Give all permutations of 3 things out of {a, b, c, d} January 1, 2017 18 / 24
Permutations of k from a set of n Give all permutations of 3 things out of {a, b, c, d} abc abd acb acd adb adc bac bad bca bcd bda bdc cab cad cba cbd cda cdb dab dac dba dbc dca dcb Would you want to do this for 7 from a set of 10? January 1, 2017 19 / 24
Combinations Choosing subsets order doesnt matter. How many ways can you do it? January 1, 2017 20 / 24
Combinations of k from a set of n Give all combinations of 3 things out of {a, b, c, d} Answer: {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d} January 1, 2017 21 / 24
Permutations and Combinations abc abd acd bcd acb adb adc bdc bac bad cad cbd bca bda cda cdb cab dab dac dbc cba dba dca dcb Permutations: 4P3 鐃 4 3 鐃 = 4C3 = 4P3 3! {a, b, c} {a, b, d} {a, c, d} {b, c, d} Combinations: 4 3 = 4C3 January 1, 2017 22 / 24
Board Question (a) Count the number of ways to get exactly 3 heads in 10 鍖ips of a coin. (b) For a fair coin, what is the probability of exactly 3 heads in 10 鍖ips? January 1, 2017 23 / 24
MIT OpenCourseWare https://ocw.mit.edu 18.05 Introduction to Probability and Statistics Spring 2014 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms.

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Introduction to probability and Statistics

  • 1. Welcome to 18.05 Introduction to Probability and Statistics Spring 2014 http://xkcd.com/904/ January 1, 2017 1 / 24
  • 2. R Free open source package. Very easy to use and install. Instructions and a link for this are on MITx/18.05r. January 1, 2017 2 / 24
  • 4. Probability vs. Statistics Di鍖erent subjects: both about random processes Probability Logically self-contained A few rules for computing probabilities One correct answer Statistics Messier and more of an art Get experimental data and try to draw probabilistic conclusions No single correct answer January 1, 2017 4 / 24
  • 5. Counting: Motivating Examples What is the probability of getting exactly 1 heads in 3 tosses of a fair coin? January 1, 2017 5 / 24
  • 6. Poker Hands Deck of 52 cards 13 ranks: 2, 3, . . . , 9, 10, J, Q, K, A 4 suits: , , , , Poker hands Consists of 5 cards A one-pair hand consists of two cards having one rank and the remaining three cards having three other rank Example: {2, 2, 5, 8, K} The probability of a one-pair hand is: (1) less than 5% (2) between 5% and 10% (3) between 10% and 20% (4) between 20% and 40% (5) greater than 40% January 1, 2017 6 / 24
  • 7. Sets in Words Old New England rule: dont eat clams (or any shell鍖sh) in months without an r in their name. S = all months L = the month has 31 days R = the month has an r in its name S = {Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec} L = {Jan, Mar, May, Jul, Aug, Oct, Dec} R = {Jan, Feb, Mar, Apr, Sep, Oct, Nov, Dec} L R = {Jan, Mar, Oct, Dec} = months with 31 days and an r January 1, 2017 7 / 24
  • 8. Visualize Set Operations with Venn Diagrams S L R L R L R Lc L R January 1, 2017 8 / 24
  • 9. Product of Sets S T = {(s, t)} January 1, 2017 9 / 24
  • 10. Inclusion-Exclusion Principle S AB A B January 1, 2017 10 / 24
  • 11. Board Question A band consists of singers and guitar players. 7 people sing 4 play guitar 2 do both How many people are in the band? January 1, 2017 11 / 24
  • 12. Rule of Product 3 shirts, 4 pants = 12 out鍖ts (More powerful than it seems.) January 1, 2017 12 / 24
  • 13. Concept Question: DNA DNA is made of sequences of nucleotides: A, C, G, T. How many DNA sequences of length 3 are there? (i) 12 (ii) 24 (iii) 64 (iv) 81 How many DNA sequences of length 3 are there with no repeats? (i) 12 (ii) 24 (iii) 64 (iv) 81 January 1, 2017 13 / 24
  • 14. Board Question 1 There are 5 Competitors in 100m 鍖nal. How many ways can gold, silver, and bronze be awarded? January 1, 2017 14 / 24
  • 15. Board Question 2 I wont wear green and red together; I think black or denim goes with anything; Here is my wardrobe. Shirts: 3B, 3R, 2G; sweaters 1B, 2R, 1G; pants 2D,2B. How many di鍖erent out鍖ts can I wear? January 1, 2017 15 / 24
  • 16. Solution answer: Suppose we choose shirts 鍖rst. Depending on whether we choose red compatible or green compatible shirts there are di鍖erent numbers of sweaters we can choose next. So we split the problem up before using the rule of product. A multiplication tree is an easy way to present the answer. R B G R,B R,B,G B,G B, D B, D B, D 3 3 2 3 4 2 4 4 4 Shirts Sweaters Pants Multiplying down the paths of the tree: Number of out鍖ts = (3 3 4) + (3 4 4) + (2 2 4) = 100 January 1, 2017 16 / 24
  • 17. Permutations Lining things up. How many ways can you do it? abc and cab are di鍖erent permutations of {a, b, c} January 1, 2017 17 / 24
  • 18. Permutations of k from a set of n Give all permutations of 3 things out of {a, b, c, d} January 1, 2017 18 / 24
  • 19. Permutations of k from a set of n Give all permutations of 3 things out of {a, b, c, d} abc abd acb acd adb adc bac bad bca bcd bda bdc cab cad cba cbd cda cdb dab dac dba dbc dca dcb Would you want to do this for 7 from a set of 10? January 1, 2017 19 / 24
  • 20. Combinations Choosing subsets order doesnt matter. How many ways can you do it? January 1, 2017 20 / 24
  • 21. Combinations of k from a set of n Give all combinations of 3 things out of {a, b, c, d} Answer: {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d} January 1, 2017 21 / 24
  • 23. Board Question (a) Count the number of ways to get exactly 3 heads in 10 鍖ips of a coin. (b) For a fair coin, what is the probability of exactly 3 heads in 10 鍖ips? January 1, 2017 23 / 24
  • 24. MIT OpenCourseWare https://ocw.mit.edu 18.05 Introduction to Probability and Statistics Spring 2014 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms.