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6.2 Experiments, Outcomes, and Events

6.2 Experiments, Outcomes, and Events. Experiment, Trial and Outcome Sample Space Event Special Events Events As Sets Mutually Exclusive Events. Experiment, Trial and Outcome.

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6.2 Experiments, Outcomes, and Events

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  1. 6.2 Experiments, Outcomes, and Events • Experiment, Trial and Outcome • Sample Space • Event • Special Events • Events As Sets • Mutually Exclusive Events

  2. Experiment, Trial and Outcome An experiment is an activity with an observable outcome. Each repetition of the experiment is called a trial. In each trial we observe the outcome of the experiment.

  3. Example Experiment, Trial and Outcome Experiment 1: Flip a coin Trial: One coin flip Outcome: Heads Experiment 2: Allow a conditioned rat to run a maze containing three possible paths Trial: One run Outcome: Path 1 Experiment 3: Tabulate the amount of rainfall in New York, NY in a year Trial: One year Outcome: 37.23 in.

  4. Sample Space The set of all possible outcomes of an experiment is called the sample space of the experiment. So each outcome is an element of the sample space.

  5. Example Sample Space • An experiment consists of throwing two dice, one red and one green, and observing the numbers on the uppermost face on each. What is the sample space S of this experiment? • Each outcome of the experiment can be regarded as an ordered pair of numbers, the first representing the number on the red die and the second number on the green die.

  6. Example Sample Space (2) • S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6) (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) (3,1), (3,2), (3,3), (3,4), (3,5), (3,6) (4,1), (4,2), (4,3), (4,4), (4,5), (4,6) (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

  7. Event • An event E is a subset of the sample space. We say that the event occurs when the outcome of the experiment is an element of E.

  8. Example Event • For the experiment of rolling two dice, describe the events • E1 = {The sum of the numbers is greater than 9}; • E2 = {The sum of the numbers is 7 or 11}. • E1 = {(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)} • E2 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (5,6), • (6,5)}

  9. Special Events Let S be the sample space of an experiment. The event corresponding to the empty set, , is called the impossible event, since it can never occur. The event corresponding to the sample space itself, S, is called the certain event because the outcome must be in S.

  10. Events As Sets Let E and F be two events of the sample space S. The event where either E or F or both occurs is designated by EF. The event where both E and F occurs is designated by EF. The event where E does not occur is designated by E '.

  11. Example Events As Sets For the experiment of rolling two dice, let E1 = “The sum of the numbers is greater than 9” and E3 = “The numbers on the two dice are equal”. Determine the sets E1E3, E1E3, and (E1E3)'.

  12. Example Events As Sets (2) E1 = {(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)} E3 = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} E1E3 = {(1,1), (2,2), (3,3), (4,4),(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)} E1E3 = {(5,5), (6,6)} (E1E3)'= {(1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,5), (5,1), (5,2), (5,3), (5,4), (6,1), (6,2), (6,3)}

  13. Mutually Exclusive Events Let E and F be events in a sample space S. Then E and F are mutually exclusive (or disjoint) if EF = . If E and F are mutually exclusive, then E and F cannot simultaneously occur; if E occurs, then F does not; and if F occurs, then E does not.

  14. Example Mutually Exclusive Events For the experiment of rolling two dice, which of the following events are mutually exclusive? E1 = “The sum of the dots is greater than 9” E2 = “The sum of the dots is 7 or 11” E3 = “The dots on the two dice are equal”

  15. Example Mutually Exclusive Events (2) E1 = {(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)} E2 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (5,6), (6,5)} E3 = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} E1E2 = {(5,6), (6,5)} - Not mutually exclusive E1E3 = {(5,5), (6,6)} - Not mutually exclusive E2E3 = - Mutually exclusive

  16. Summary Section 6.2 - Part 1 • The sample space of an experiment is the set of all possible outcomes of the experiment. Each subset of the sample space is called an event. We say that an event occurs when the outcome is an element of the event. • The event EF occurs when either E or F or both occurs. The event EF occurs when both E and F occur. The event E' occurs when E does not occur.

  17. Summary Section 6.2 - Part 2 • Two events are mutually exclusive if they cannot both occur at the same time.

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