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Basic Probability

Basic Probability. Chapter 7B. What is Probability?. Probability is the chance that something is the case or will happen Example: There are 5 color balls in a bag, 3 of them are red and 2 of them are blue. If we arbitrarily pick 2 balls out, what is the chance to pick the same color balls?.

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Basic Probability

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  1. Basic Probability Chapter 7B

  2. What is Probability? Probability is the chance that something is the case or will happen Example: There are 5 color balls in a bag, 3 of them are red and 2 of them are blue. If we arbitrarily pick 2 balls out, what is the chance to pick the same color balls?

  3. Let us mark 5 color balls with number 1-5 as the following: Then all possible outcomes for picking 2 balls are:

  4. Answer • There are 10 possible outcomes totally. • Three of them are both reds, One of them is both blues. • So the probability for picking the same color balls is to pick 4 outcomes form 10 outcomes. It is

  5. Probability is to use number to express the chance that some event in case will certain happen. 1. If some event definitely will happen, we say the chance is 100% percent. Or say its probability to happen is 1. 2. If tossing a coin, then the chances to get head or tail is half-half. So we say the chance to get head is 50%. Or say that the probability to get head is 0.5. 3. If some event definitely will not happen, then we say its chance is zero. Or say its probability to happen is 0. 4. If the probability is greater than 0.5, we will say it is likely to happen. Otherwise, we will say it will be unlikely to happen.

  6. Sample Space In the above examples, picking two balls, tossing coins are called probability experiment trials. The result of each trial is called experiment outcome. Definition: All possible different outcomes of a probability experiment is called a sample space. Basic properties of sample space 1. Everyoutcome of a sample space has the same chance to happen. 2. Two different outcomes of a sample space never happen together.

  7. Event & its Probability Definition: A subset E of outcomes is called an event E. Probability of Event: the basic formula is Events of same color Sample space

  8. Some basic Probability Rules Rule 1: The probability of any event E is a number between and including 0 and 1. 0 ≤ P(E) ≤ 1 Rule 2: If an event E cannot occur, which means the event E is not in the sample space, then its probability is 0. P(E) = 0 Rule 3: If an event E is certainly occur, then its probability is 1. P(E) = 1 Rule 4: Let a sample space consist of n distinct events s1, s2, s3, . . ., sn. Then the sum of the probabilities of all those events is 1 P(s1) +P(s2) +P(s3) +. . . +P(sn) = 1

  9. Sample space examples

  10. Sample space for rolling two dice What is the probability that the sum of two dice number is 8?

  11. Tossing a coin When tossing a coin, there are only two outcomes, head or tail. Because the chance to get head or tail is equal, so the probability to get head is 1/2 = 0.5

  12. Tossing two coins When tossing two coins, what is the probability to get one head and one tail? To solve this question , we design outcomes into two events: either both same or different. We can feel that both events will have the equal chance to happen. Therefore, the probability to get one head and one tail is 50% or 0.5.

  13. Tossing outcomes tree We try to use outcomes tree to solve the above question. We mark two coins as coin1 and coin2. Then put the outcomes of coin1 in the first line and the outcomes of the coin2 in the second line. Then we get the following outcomes tree H T coin1 coin2 HH HT TH TT There are 4 outcomes, so the probability is 2/4 = 0.5

  14. Tossing three coins When tossing three coins, what is the probability to get two heads and one tail? We draw outcome tree coin1 H T coin2 T H T H coin3 H T H T H T H T HHH HHT HTH HTT THH THT TTH TTT There are 8 outcomes, so the probability is 3/8 = 0.375

  15. Exercise When tossing four coins, what is the probability to get two heads and two tails? draw outcome tree first.

  16. Rolling a die Question 1: What is probability to get number 3 for rolling a die? Solution: Totally we have 6 outcomes and get number 3 is one of them. So the probability is 1/6 = 0.167. Question 2: What is probability to get number less then 7 for rolling a die? Solution: All numbers of outcomes are less than 7. Therefore the probability to get number less then 7 is 1. Question 3: What is probability to get number 9 for rolling a die? Solution: All number of a die is from 1 to 6. So we never could get 9. Therefore the probability to get number 9 is zero.

  17. Drawing card A regular deck of cards has 52 cards of four types as the following

  18. Drawing one card

  19. Addition Event Definition 2: Let E and F be two events. Then the event that E or F occurs is called the addition of event E and F, which means at least one event occurs, denoted as EF. F E Note: Event EF has the following 3 cases: 1. Event E occurs, event F not occurs. 2. Event E occurs, event F not occurs. 3. Both both E and F occur.

  20. Exclusive Events and addition rule Definition 1: Let E and F be two events. If they never occur at the same time, then E and F be two events are called the exclusive events. For example: When rolling a die, the event to get number 2 and the event to get number 3 are exclusive events. Theorem. Let E and F be two events be two exclusive events. Then P(EF)= P(E)+ P(F) This is called probability addition rule.

  21. Multiplication Event Definition 1: Let E and F be two events. Then the event that E and F both occur is called the multiplication of event E and F, denoted as EF. F E Example: Drawing one card from a regular deck, let event E be to draw a spade card and event F be to draw a card of number 4. Then the event of EF is to draw a spade 4 card.

  22. Independent Events and Multiplicative Rule Definition 1: Let E and F be two events. If one event occurs or not has no relation with whether other event occur or not, then events E and F are called the independent events. For example: A family has two children. Let E be the event that the first child is a boy. Let F be the event that the second child is a girl. Then E and F are the independent events Theorem. Let E and F be two events be two independent events. Then P(EF)= P(E)P(F) This is called probability multiplicative rule.

  23. Complement Event Definition 3: Let E be an events. Then the event that E does not occurs is called the complement event of E, denoted as E' E Example : Rolling a die, let E be the event to get number 3. Then E' is the event to get a number other than 3.

  24. Complementary Rule Theorem. Let E an events Then P(E' ) =1-P(E) This is called probability complementary rule.

  25. Odds

  26. Re-solve two coins tossing When tossing two coins, what is the probability to get one head and one tail? Event B Event A

  27. Re-solve three coins tossing When tossing three coins, what is the probability to get two heads and one tail?

  28. Conditional Probability Let E and F be two events. Then the probability for E occurs after F occurs is called the conditional event written as F|E. E F E F Theorem:

  29. Example The probability that Sam parks in a no-parking zone and get a parking ticket is 0.06. And the probability that Sam cannot find a legal parking lot and has to parking in a no-parking zone is 0.20. Today, Sam parks his car in a a no-parking zone. Find the probability that Sam will get a ticket.

  30. An important case

  31. Three coins tossing H T T H T H H T H T H T H T HHH HHT HTH HTT THH THT TTH TTT

  32. Drawing Cards

  33. Exercises

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