CHAPTER 8_A PROBABILITY MODELS BERNOULLI TRIAL

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Presentation transcript:

CHAPTER 8_A PROBABILITY MODELS BERNOULLI TRIAL A RANDOM EXPERIMENT WITH TWO COMPLEMENTARY EVENTS, SUCCESS (S) AND FAILURE (F) IS CALLED A BERNOULLI TRIAL. P(SUCCESS) = p P(FAILURE) = q = 1 - p

SUCCESS = HEADS WITH p = 0.5 AND FAILURE = TAILS WITH q = 1 – p = 0.5 EXAMPLES TOSSING A COIN SUCCESS = HEADS WITH p = 0.5 AND FAILURE = TAILS WITH q = 1 – p = 0.5 TAKING A MULTIPLE CHOICE EXAM UNPREPARED. SUCCESS = CORRECT ANSWER FAILURE = WRONG ANSWER p = 0.2; q = 1 – p = 1 – 0.2 = 0.8

PRODUCTS COMING OUT OF A PRODUCTION LINE SUCCESS = DEFECTIVE ITEMS FAILURE = NON-DEFECTIVE ITEMS ROLLING A DIE 10 TIMES SUCCESS = GETTING A 6; p = 1/6 FAILURE = NOT GETTING A 6; q = 5/6

CONDITIONS THE FOLLOWING CONDITIONS MUST HOLD BEFORE USING THE BINOMIAL PROBABILITY MODEL. (1) THE TRIALS MUST BE BERNOULLI, THAT IS, THE RANDOM EXPERIMENT MUST HAVE TWO COMPLEMENTARY OUTCOMES – SUCCESS AND FAILURE; (2) THE TRIALS MUST BE INDEPENDENT OF ONE ANOTHER; (3) THE PROBABILITY OF SUCCESS IS THE SAME FOR EACH TRIAL. (4) THE NUMBER OF TRIALS n, MUST BE FIXED.

BINOMIAL PROBABILITY MODEL FOR BERNOULLI TRIALS QUESTION: WHAT IS THE NUMBER OF SUCCESSES IN A SPECIFIED NUMBER OF TRIALS? THE BINOMIAL PROBABILITY MODEL ANSWERS THIS QUESTION, THAT IS, THE PROBABILITY OF EXACTLY k SUCCESSES IN n TRIALS.

BINOMIAL PROBABILITY MODEL LET n = NUMBER OF TRIALS p = PROBABILITY OF SUCCESS q = PROBABILITY OF FAILURE X = NUMBER OF SUCCESSESS IN n TRIALS

n! = n(n-1)(n-2)(n-3) … 3.2.1

EXAMPLES COMPUTE (1) 3! (2) 4! (3) 5! (4) 6!

EXAMPLE ASSUME THAT 13% OF PEOPLE ARE LEFT-HANDED. IF WE SELECT 5 PEOPLE AT RANDOM, FIND THE PROBABILITY OF EACH OUTCOME BELOW. (1) THERE ARE EXACTLY 3 LEFTIES IN THE GROUP. 0.0166 (2) THERE ARE AT LEAST 3 LEFTIES IN THE GROUP. 0.0179 (3) THERE ARE NO MORE THAN 3 LEFTIES IN THE GROUP. 0.9987

EXAMPLE AN OLYMPIC ARCHER IS ABLE TO HIT THE BULL’S-EYE 80% OF THE TIME. ASSUME EACH SHOT IS INDEPENDENT OF THE OTHERS. IF SHE SHOOTS 6 ARROWS, WHAT’S THE PROBABILITY THAT (1) SHE GETS EXACTLY 4 BULL’S-EYES? 0.246 (2) SHE GETS AT LEAST 4 BULL’S-EYES? 0.901 (3) SHE GETS AT MOST 4 BULL’S-EYES? 0.345 (4) SHE MISSES THE BULL’S-EYE AT LEAST ONCE? 0.738 (5) HOW MANY BULL’S-EYES DO YOU EXPECT HER TO GET? 4.8 BULL’SEYES (6) WITH WHAT STANDARD DEVIATION? 0.98