Problems3

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Problems3

  1. 1. Problems Probability-3
  2. 2. :3/4 Probability:   {yellow, blue, green, red} Sample Space:   A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on a sector that is not red after spinning this spinner? Question No 1
  3. 3. Question No 2: <ul><li>Experiment 2:   A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a card that is not a king? </li></ul><ul><li>12/13 </li></ul>
  4. 4. Question no 3 <ul><li>   A single 6-sided die is rolled. What is the probability of rolling a number that is not 4? </li></ul><ul><li>5/6 </li></ul>
  5. 5. Question No 4 <ul><li>A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a card that is a club? </li></ul><ul><li>13/52 </li></ul>
  6. 6. Question no 5 <ul><li>A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die </li></ul><ul><li>1/12 </li></ul>
  7. 7. Question no 6 <ul><li>A card is chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and an eight? </li></ul><ul><li>   1   </li></ul><ul><li>169 </li></ul>  
  8. 8. Addition Rule <ul><li>The addition rule is a result used to determine the probability that event A or event B occurs or both occur. </li></ul><ul><li>Non mutually exclusive </li></ul><ul><li>The result is often written as follows, using set notation: </li></ul><ul><li>Where P(A) = probability that event A occurs </li></ul><ul><li>P(B) = probability that event B occurs </li></ul><ul><li>= probability that event A or event B occurs </li></ul><ul><li>= probability that event A and event B both occur </li></ul><ul><li>For mutually exclusive events , that is events which cannot occur together: </li></ul><ul><li>= 0 </li></ul>
  9. 9. Addition Rule Contd…. <ul><li>The addition rule therefore reduces to </li></ul><ul><li>= P(A) + P(B) </li></ul><ul><li>For independent events , that is events which have no influence on each other </li></ul>The addition rule therefore reduces to
  10. 10. Multiplication Rule <ul><li>The multiplication rule is a result used to determine the probability that two events, A and B, both occur. </li></ul><ul><li>The multiplication rule follows from the definition of conditional probability. </li></ul><ul><li>The result is often written as follows, using set notation: </li></ul><ul><li>where: P(A) = probability that event A occurs </li></ul><ul><li>P(B) = probability that event B occurs </li></ul><ul><li>= probability that event A and event B occur </li></ul><ul><li>P(A | B) = the conditional probability that event A occurs given that event B has occurred already </li></ul><ul><li>P(B | A) = the conditional probability that event B occurs given that event A has occurred already </li></ul><ul><li>For independent events, that is events which have no influence on one another, the rule simplifies to </li></ul><ul><li>:That is, the probability of the joint events A and B is equal to the product of the individual probabilities for the two events. </li></ul>
  11. 11. <ul><li>Statistical Independence </li></ul><ul><li>When the occurrence of one event does not affect and is not affected by the probability of occurrence of any other event </li></ul><ul><li>Probabilities under statistical independence – </li></ul><ul><li>Marginal Probability </li></ul><ul><li>Joint probability </li></ul><ul><li>Conditional Probability </li></ul>
  12. 12. Marginal Probability <ul><li>Refers one and only one event </li></ul>
  13. 13. Joint Probability <ul><li>The Probability of two or more independent events occur together or in succession is called joint probability </li></ul><ul><li>P(AB)=P(A  B)=P(A)*P(B) </li></ul>
  14. 14. Conditional Probability <ul><li>For statistically independent events A and B the conditional probability denoted by P(A/B) of event A given that B has already occurred is simply the probability of event A </li></ul><ul><li>P(A/B)=P(A) </li></ul><ul><li>Like that </li></ul><ul><li>P(B/A)=P(B) </li></ul>
  15. 15. Statistical Dependence events <ul><li>Marginal Probability </li></ul><ul><li>Same like statistical Independence </li></ul><ul><li>P(A)= P(A  B)+ P(A  B) </li></ul><ul><li>P(B)= P(A  B)+ P(A  B) </li></ul>
  16. 16. Joint Probability <ul><li>Dependent Events </li></ul><ul><li>P(A and B) ≠P (A  B) ≠ P(A  )*P(B) </li></ul><ul><li>P(A) ≠ P(A/B) and P(B) ≠ P(B/A) </li></ul><ul><li>The Probability of two or more dependent events occur together or in succession is called joint probability </li></ul><ul><li>P(A  B)=P(A)*P(B/A) </li></ul><ul><li>P(A  B)=P(B)*P(A/B) </li></ul>
  17. 17. Conditional Probability <ul><li>P(A  B) </li></ul><ul><li>P(A/B) = </li></ul><ul><li>P(B) </li></ul><ul><li>Gives Conditional Probability of B given that the event A has occurred </li></ul>
  18. 18. Baye’s Theorem <ul><li>Suppose an event has happened as a result of several causes . Then we are interested to find out the probability of a particular cause which really affected the event to happen. Problems of this type are called inverse probability . Baye’s theorem is based on inverse probability </li></ul>

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