- 3. Introduction to Counting and Probability Some Terms “Probability is the branch of mathematics that provide quantitative description of the likely occurrence of an event. Outcome – any possible result of an experiment or operation Sample space – the complete list of all possible outcomes of an experiment or operation Event – refers to any subset of a sample space Counting – operation used to find the number of possible outcomes
- 4. Introduction to Counting and Probability Counting Problems Counting problems are of the following kind: “How many combinations can I make with 5 Tshirts, 4 pairs of pants, and 3 kinds of shoes? “How many ways are there to pick starting 5 players out of a 12-player basketball team?” Most importantly, counting is the basis for computing probabilities. Example;:“What is the probability of winning the lotto?”
- 5. Introduction to Counting and Probability Counting Problems Example: Ang Carinderia ni Jay ay may breakfast promo kung saan maaari kang makabuo ng combo meal mula sa mga sumusunod: SILOG DRINKS DESSERT TAPSILOG KAPE SAGING TOSILOG MILO BROWNIES LONGSILOG ICED TEA BANGSILOG
- 6. Introduction to Counting and Probability Counting Problems Example: Question: If you want to create a combo meal by choose one of each kind, how many choices can you have? SILOG DRINKS DESSERT TAPSILOG KAPE SAGING TOSILOG MILO BROWNIES LONGSILOG ICED TEA BANGSILOG
- 7. Introduction to Counting and Probability Counting Problems SILOG DRINKS DESSERT TAPSILOG KAPE SAGING TOSILOG MILO BROWNIES LONGSILOG ICED TEA BANGSILOG KAPE TAPSILOG SAGING BROWNIES MILO SAGING BROWNIES ICED TEA SAGING BROWNIES
- 8. Introduction to Counting and Probability Counting Problems SILOG DRINKS DESSERT TAPSILOG KAPE SAGING TOSILOG MILO BROWNIES LONGSILOG ICED TEA BANGSILOG KAPE TOSILOG SAGING BROWNIES MILO SAGING BROWNIES ICED TEA SAGING BROWNIES
- 9. Introduction to Counting and Probability Counting Problems SILOG DRINKS DESSERT TAPSILOG KAPE SAGING TOSILOG MILO BROWNIES LONGSILOG ICED TEA BANGSILOG KAPE LONGSILOG SAGING BROWNIES MILO SAGING BROWNIES ICED TEA SAGING BROWNIES
- 10. Introduction to Counting and Probability Counting Problems SILOG DRINKS DESSERT TAPSILOG KAPE SAGING TOSILOG MILO BROWNIES LONGSILOG ICED TEA BANGSILOG KAPE BANGSILOG SAGING BROWNIES MILO SAGING BROWNIES ICED TEA SAGING BROWNIES
- 11. Fundamental Principle of Counting TOSILOG TAPSILOG SAGING KAPE SAGING KAPE BROWNIES BROWNIES SAGING MILO BROWNIES SAGING MILO BROWNIES SAGING ICED TEA SAGING ICED TEA BROWNIES BROWNIES LONGSILOG BANGSILOG SAGING KAPE SAGING KAPE BROWNIES BROWNIES SAGING MILO BROWNIES SAGING MILO SAGING ICED TEA BROWNIES SAGING ICED TEA BROWNIES There are 24 possible combo meals BROWNIES
- 12. Fundamental Principle of Counting The Product Rules The Product Rule: Suppose that a procedure can be broken down into two successive tasks. If there are n1 ways to do the first task and n2 ways to do the second task after the first task has been done, then there are n1n2 ways to do the procedure.
- 13. Introduction to Counting and Probability The Product Rules Generalized product rule: If we have a procedure consisting of sequential tasks T1, T2, …, Tn that can be done in k1, k2, …, kn ways, respectively, then there are n1 n2 … nm ways to carry out the procedure.
- 14. Introduction to Counting and Probability Basic Counting Principles The two product rules are collectively called the FUNDAMENTAL PRINCIPLE OF COUNTING Woohoo…
- 15. Introduction to Counting and Probability The Product Rules Generalized product rule: If we have a procedure consisting of sequential tasks T1, T2, …, Tm that can be done in k1, k2, …, kn ways, respectively, then there are n1 n2 … nm ways to carry out the procedure. T1 k1 T2 k2 T3 k3 … … Tn Tasks kn No. of ways no. of outcomes in all = k1k2k3 ...kn
- 16. Introduction to Counting and Probability Basic Counting Principles Example 1 If you have 5 T-shirts, 4 pairs of pants, and 3 pairs of shoes, how many ways can you choose to wear three of them? Solution The number of ways is 5 4 3 60
- 17. Introduction to Counting and Probability Basic Counting Principles Example 2 How many outcomes can you have when you toss: 2 (head and tail) a. One coin? 2x2=4 b. Two coins? 2x2x2=8 c. Three coins?
- 18. Introduction to Counting and Probability Basic Counting Principles Example 3 How many outcomes can you have when you toss: 6 a. One die? 6 x 6 = 36 b. Two dice? 6 x 6 x 6 = 216 c. Three dice? 6 x 2 = 12 d. A die and a coin?
- 19. Fundamental Principle of Counting Basic Counting Principles Example 4 How many ways can you answer a a. 20-item true or false quiz? 20 x 2 = 40 b. 20-item multiple choice test, with choices A, B, C, D? 20 x 4 = 80
- 20. Fundamental Principle of Counting Basic Counting Principles Example 5 How many three-digit numbers can form from the digits 1, 2, 3, 4 if the digits a. can be repeated? 4 x 4 x 4 = 64 b. cannot be repeated? 4 x 3 x 2 = 24
- 21. Fundamental Principle of Counting Basic Counting Principles Example 7 How many three-digit EVEN numbers can form from the digits 1, 2, 3, 4 if the digits a. can be repeated? 4 x 4 x 2 = 32 b. cannot be repeated? 3 x 3 x 2 = 12
- 22. Introduction to Counting and Probability Basic Counting Principles Example 8 How many three-digit numbers can form from the digits 0,1, 2, 3, 4 if the digits a. can be repeated? 4 x 5 x 5 = 100 b. cannot be repeated? 4 x 4 x 3 = 48
- 23. Introduction to Counting and Probability Check your understanding 1. 2. 3. How many subdivision house numbers can be issued using 1 letter and 3 digits? How many ways can you choose one each from 10 teachers, 7 staff, and 20 students to go to an out-of-school meeting? How many 4-digit numbers can be formed from the digits 1 2, 3, 4, 5 if the digits cannot be repeated? Answers: 1. 26 x 10 x 10 x 10 = 26,000 2. 10 x 7 x 20 = 1,400 3. 5 x 4 x 3 x 2 = 120
- 24. Introduction to Counting and Probability Basic Probability Probability is a relative measure of expectation or chance that an event will occur. Question: How likely is an event to occur based on all the possible outcomes?
- 25. Introduction to Counting and Probability Basic Probability Computing Probability The probability p than an event can occur is the ratio of the number of ways that the event will occur over the number of possible outcomes S. number of ways that a certain event will occur p number of possible outcomes
- 26. Introduction to Counting and Probability Basic Probability Example 1 There are two outcomes in a toss of a coin – head or tail. Thus, the probability that a head will turn up in a coin toss is 1 out of 2; that is, number of heads 1 p number of possible outcomes 2
- 27. Introduction to Counting and Probability Basic Probability Example 2 There are 6 outcomes in a roll of die. What is the probability of getting a a. 6? One out of 6: p = 1/6 b. 2 or 3? Two out of 6: p = 2/6 or 1/3 c. odd number? Three out of 6: p = 3/6 or 1/2 Zero out of 6: p = 0/6 or 0 d. 8? Letter d is an IMPOSSIBLE EVENT
- 28. Introduction to Counting and Probability Basic Probability Example 3 A bowl has 5 blue balls, 6 red balls, and 4 green balls. If you draw a ball at random, what is the probability that you’ll 5 out of 15: p = 5/15 = 1/3 a. get a blue ball? 6 out of 15: p = 6/15 = 2/5 b. get a red ball? 4 out of 15: p = 4/15 c. a green ball? d. not get a red ball? 5 + 4 = 9 out of 15: p = 9/15 or 3/5 e. get a red or green ball? 6 + 4 = 10 out of 15: p = 10/15 = 2/3
- 29. Introduction to Counting and Probability Check your understanding 1. 2. 3. Two coins are tossed. What is the probability of getting two tails? In a game of Bingo, what is the probability that the first ball comes from the letter G? All the three-digit numbers formed by using the digits 1, 2, 3, and 4 without repeating digits are put in a bowl. What is the probability that when a number is drawn, it is odd?
- 30. Introduction to Counting and Probability Check your understanding 1. 1 out of 8: p = 1/8 2. 10 out of 75: p = 10/75 = 2/15 3. Number of 3-digit numbers: 4 x 3x2 = 24 Number of odd 3-digit numbers: 3 x 2 x 2 = 12 p = 12/24 = 1/2