1. 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

2. 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

3. 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

4. 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.

5. 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.

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

7. n! = n(n-1)(n-2)(n-3) … 3.2.1

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

9. 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

10. 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? BULL’SEYES
(6) WITH WHAT STANDARD DEVIATION? 0.98