PROBABILITY union and intersection of events.pptx

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1. PROBABILITY

2. probability Probability  is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true

3. EXPERIMENT an  experiment  or trial is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes

4. outcome an  outcome  is a possible result of an experiment or trial

5. Sample space The  sample space  of a random experiment is the collection of all possible outcomes

6. EVENT an  event  is an outcome or defined collection of outcomes of a random experiment. Since the collection of all possible outcomes to a random experiment is called the sample space, another defIniton of  event  is any subset of a sample space.

7. Probability line

8. FORMULA: P ( event ) =  

9. Example: IN THE EXPERIMENT OF TOSSING A COIN , WHAT IS THE PROBABILITY THAT THE COIN WILL land with tail in front? SAMPLE SPACE = (TAIL, HEAD) OUTCOMES DISIRED EVENT = GETTING A TAIL P ( TAIL ) = or 50%   EVEN CHANCE

10. coin

11. DICE

12. PLAYING CARDS

13. Example: IN THE EXPERIMENT OF ROLLING A DIE, WHAT IS THE PROBABILITY OF GETTING NUMBER 5 ? SAMPLE SPACE = (1, 2, 3, 4, 5, 6) OUTCOMES DISIRED EVENT = GETTING NUMBER 5 P ( 5 ) = or 17%   UNLIKELY

14. Example: IN THE EXPERIMENT OF ROLLING A DIE, WHAT IS THE PROBABILITY OF GETTING an even number? SAMPLE SPACE = (1, 2, 3, 4, 5, 6) OUTCOMES DISIRED EVENT = GETTING AN EVEN # P ( EVEN ) = or 50%   EVEN CHANCE

15. Example: IN THE EXPERIMENT OF ROLLING A DIE, WHAT IS THE PROBABILITY OF GETTING A NUMBER LESS THAN 5? SAMPLE SPACE = (1, 2, 3, 4, 5, 6) OUTCOMES DISIRED EVENT = GETTING A # LESS THAN 5 P (#< 5 ) = or 67%   LIKELY

16. Example: IN THE EXPERIMENT OF ROLLING A DIE, WHAT IS THE PROBABILITY OF GETTING A NUMBER GREATER THAN 6? SAMPLE SPACE = (1, 2, 3, 4, 5, 6) OUTCOMES DISIRED EVENT = GETTING A # GREATER THAN 6 P (#> 5 ) = or 0%   IMPOSSIBLE

17. Example: IN THE EXPERIMENT OF PICKING A CARD FROM A DECK OF PLAYING CARDS, WHAT IS THE PROBABILITY OF GETTING A RED CARD? SAMPLE SPACE = (52…) OUTCOMES DISIRED EVENT = GETTING A RED CARD P ( RED ) = or 50%   EVEN CHANCE

18. Example: IN THE EXPERIMENT OF PICKING A CARD FROM A DECK OF PLAYING CARDS, WHAT IS THE PROBABILITY OF GETTING A SPADE CARD? SAMPLE SPACE = (52…) OUTCOMES DISIRED EVENT = GETTING A SPADE CARD P ( SPADE ) = or 25%   UNLIKELY

19. Example: IN THE EXPERIMENT OF PICKING A CARD FROM A DECK OF PLAYING CARDS, WHAT IS THE PROBABILITY OF GETTING A FACE CARD? SAMPLE SPACE = (52…) OUTCOMES DISIRED EVENT = GETTING A FACE CARD P ( FACE ) = or 23%   UNLIKELY

20. Example: IN THE EXPERIMENT OF PICKING A CARD FROM A DECK OF PLAYING CARDS, WHAT IS THE PROBABILITY OF GETTING A NUMBER CARD? SAMPLE SPACE = (52…) OUTCOMES DISIRED EVENT = GETTING A NUMBER CARD P ( NUMBER ) = or 77%   LIKELY

21. UNION AND INTERSECTION OF EVENTS

22. Union of events The union of two events A and B denoted by A U B , is obtained by combining the outcomes of events A and B.

23. INTERSECTION of events The intersection of two events A and B denoted by A Ƞ B , is obtained by grouping common events of all outcomes of A and B.

24. Mutually exclusive events Events that do not have a common elements. Not Mutually exclusive events Events that do have a common elements.

25. Probability of mutually exclusive events If events A and B are mutually exclusive events, then P(A U B) = P(A) + P(B)

26. Example: One die is rolled. Find the probability the probability of getting number 4 or 6. SAMPLE SPACE = (1, 2, 3, 4, 5, 6) OUTCOMES EVENT A = GETTING NUMBER 4 P ( A ) =   EVENT B = GETTING NUMBER 6 P ( B ) =   P ( A U B ) = + =  

27. Probability of NOT mutually exclusive events If events A and B are NOT mutually exclusive events, then P(A U B) = P(A) + P(B) – P(A Ƞ B)

28. Example: One die is rolled. Find the probability the probability of getting number 4 OR EVEN NUMBER. SAMPLE SPACE = (1, 2, 3, 4, 5, 6) OUTCOMES EVENT A = GETTING NUMBER 4 P ( A ) =   EVENT B = GETTING EVEN NUMBER P ( B ) =  

29. Example: WHAT IS THE PROBABILITY OF GETTING AND ODD NUMBER AND A PRIME NUMBER FROM TURNING THE SPINNER BELOW? SAMPLE SPACE = (1, 2, 3, 4, 5, 6, 7, 8) OUTCOMES EVENT A = GETTING AN ODD NUMBER EVENT B = GETTING A PRIME NUMBER 1 2 3 4 5 6 7 8

30. MULTIPLICATION RULE FOR INDEPENDENT EVENTS If events A and B are independent events, the probability that both A and B will occur is P(A Ƞ B) = P(A) x P(B)

31. Example: What is the probability of getting a six in the first roll of a die and one in the second roll? EVENT A = GETTING NUMBER SIX EVENT B = GETTING NUMBER ONE

32. Example: A JAR CONTAINS FOUR RED BALLS AND FOUR BLUE BALLS . WHAT IS THE PROBABILITY OF GETTING A RED BALL FIRST AND A BLUE BALL SECOND? EVENT A = GETTING A RED BALL EVENT B = GETTING A BLUE BALL

33. Probability rule for the complement of an event If A and A’ are complementary events, then P(A) + P(A’) = 1

34. Example: The probability that jenny will pass the test on mathematics is 4/5. What is the probability that she will fail? EVENT A = JENNY WILL PASS EVENT A’ = JENNY WILL FAIL (WILL NOT PASS)

35. Example: WHAT IS THE PROBABILITY OF NOT GETTING A NUMBER 2 FROM ROLLING A DIE? EVENT A = GETTING NUMBER 2 EVENT A’ = NOT GETTING NUMBER 2

36. Answer the following probability 1. What is the probability of getting an even number and a number greater than 5 you spined the roulette? 2. Find the probability of getting an ace card or a face card when you picked a card from a deck of playing cards. 1 2 3 4 5 6 7 8

37. 1. What is the probability of getting an odd number from the first roll of a die and number 6 in the second roll? 2. In the experiment of rolling two dice, what is the probability of getting a sum of 8 from both dice.

38. 3. What is the probability of getting an even number from the first spin of the roulette and prime number in the second spin? 1 2 3 4 5 6 7 8