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Binomial & Multinomial Probability (2).pptx
- 1. Probability Distribution Binomial STEPHANY CAMILLEG. PASCUA Discussant & Multinomial
- 2. The Binomial Probability Distributionis a very good approach for resolving probability involving random experiment which has two possible outcomes. The outcome that the event (a) will occur; (b) will not occur.
- 3. The Formula: P(x) =C p q n x (n-x) Where: P(X) = probability of X n = number of trials x = number of successes among trials p = probability of success q = probability of failure x n C x = n! (n-x)! x! __n!___ (n-x)! x!
- 4. Example1: When an unbiasedcoin is tossed seven times, what is the probability of obtaining exactly 4 heads? n = 7 x = 4 p = 1/2 q = 1/2
- 5. P(x) = Cp q = (1/2) (1/2) = = 35 = or 27% n x (n-x) x n = 7 x = 4 p = 1/2 q = 1/2 4 3 __n!___ (n-x)! x! __7!___ (3)! 4! _1_ 16 _1_ 8 _1_ 16 _1_ 8 _35_ of chance for the head to appear 4 times in 7 tosses
- 6. Example 2: When anunbiased coin is tossed seven times, what is the probability of obtaining exactly 6 heads? n = 7 x = 6 p = 1/2 q = 1/2
- 7. P(x) = Cp q = (1/2) (1/2) = = 7 = or 5% n x (n-x) x n = 7 x = 4 p = 1/2 q = 1/2 6 1 __n!___ (n-x)! x! __7!___ (1)! 6! _1_ 64 _1_ 2 _1_ 64 _1_ 2 _7_ of chance for the head to appear 6 times in 7 tosses
- 8. Example 3: A teacherdeveloped a 5-item multiple choice questions with four options in each item. What is the probability that a certain student who randomly selects his answers will get exactly 4 correct answers? n = 5 x = 4 p = 1/4 q = 3/4
- 9. P(x) = Cp q = (1/4) (3/4) = = 5 = or 1% n x (n-x) x n = 5 x = 4 p = 1/4 q = 3/4 4 1 __n!___ (n-x)! x! __5!___ (1)! 4! _1_ 256 _3_ 4 _1_ 256 _3_ 4 _15_ probability to have 4 correct answers out of 5 questions with 4 options
- 10. Example 4: When anunbiased coin is tossed seven times, what is the probability of obtaining at most 3 heads? n = 7 x = 3 p = 1/2 q = 1/2 x = 2 x = 1 x = 0
- 11. P(x) = Cp q = (1/2) (1/2) = = 35 = or 27% n x (n-x) x n = 7 x = 3 p = 1/2 q = 1/2 3 4 __n!___ (n-x)! x! __7!___ (4)! 3! _1_ 8 _1_ 16 _1_ 8 _1_ 16 35_
- 12. Example 4: When anunbiased coin is tossed seven times, what is the probability of obtaining at most 3 heads? n = 7 x = 3 = p = 1/2 q = 1/2 x = 2 = x = 1 x = 0 35_ 128
- 13. P(x) = Cp q = (1/2) (1/2) = = 21 = or 16% n x (n-x) x n = 7 x = 2 p = 1/2 q = 1/2 2 5 __n!___ (n-x)! x! __7!___ (5)! 2! _1_ 4 _1_ 32 _1_ 4 _1_ 32 21_
- 14. Example 4: When anunbiased coin is tossed seven times, what is the probability of obtaining at most 3 heads? n = 7 x = 3 = p = 1/2 q = 1/2 x = 2 =....... x = 1 =....... x = 0 =....... 35_ 128 21_ 128
- 15. P(x) = Cp q = (1/2) (1/2) = = 7 = or 5% n x (n-x) x n = 7 x = 1 p = 1/2 q = 1/2 1 6 __n!___ (n-x)! x! __7!___ (6)! 1! _1_ 2 _1_ 64 _1_ 2 _1_ 64 7_
- 16. Example 4: When anunbiased coin is tossed seven times, what is the probability of obtaining at most 3 heads? n = 7 x = 3 = p = 1/2 q = 1/2 x = 2 =....... x = 1 =....... x = 0 =....... 35_ 128 21_ 128 7_ 128
- 17. P(x) = Cp q = (1/2) (1/2) = 1 = 1 1 = or 1% n x (n-x) x n = 7 x = 0 p = 1/2 q = 1/2 0 7 __n!___ (n-x)! x! __7!___ (7)! 0! _1_ 128 _1_ 128 1_
- 18. Example 4: When anunbiased coin is tossed seven times, what is the probability of obtaining at most 3 heads? n = 7 x = 3 = p = 1/2 q = 1/2 x = 2 =....... x = 1 =....... x = 0 =....... 35_ 128 21_ 128 7_ 128 1_ 128 P(X ≤ 3) = P (X=3) + P (X=2) + P (X=1) + P (X=0) = + + + = or or 50% 35_ 21_ 7_ 128 1_ 128 64_ 1 of chance for the head to appear at most 3 times in 7 tosses
- 19. The Multinomial ProbabilityDistribution has more than 2 possible outcomes. It deals with experiments where each trial results in one of K possible outcomes. If there an N independent trials and each outcome has a constant probability, we can calculate the probability of observing specific counts for each outcome.
- 20. ____n!____ x ! .x ! ... x ! P = . (p) . (p) ... (p) 1 2 k 1 2 k Likelihood of seeing a specific outcomes Ways to arrange different outcomes into different categories Where: n = total number of trials x = number of occurances for each category k = number of possible categories p = probability of success for each category
- 21. Example 1: A boxcontains 4 red, 3 blue, and 3 green balls. If 6 balls are drawn with replacement, what is the probability of getting 2 red, 2 blue, and 2 green balls? ____n!____ x ! . x ! ... x ! P = . (p) . (p) ... (p) 1 2 k 1 2 k P = . (0.4) .(0.3) .(0.3) ____6!____ 2! . 2! . 2! 2 n = 6 X1 = 2 p1 = 0.4 X2 = 2 p2 = 0.3 X3 = 2 p3 = 0.3 2 2 P = 90 x 0.16 x 0.09 x 0.09 P = 0.1166 or 11.66%
- 22. Example 2: The surveyfinds that 40% of people like chocolate, 25% like vanilla, 20% like mango, and 15% like strawberry flavor. What is the probability that out of 14 people, 5 like chocolate, 4 like vanilla , 3 like mango, and 2 like strawberry? ____n!____ x ! . x ! ... x ! P = . (p) . (p) ... (p) 1 2 k 1 2 k P = . (0.4) .(0.25) .(0.2) .(0.15) ____14!____ 5!.4!.3!.2! 5 n = 14 X1 = 5 p1 = 0.4 X2 = 4 p2 = 0.25 X3 = 3 p3 = 0.2 X4 = 2 p4 = 0.15 4 3 P = 2,522,520 x 0.01 x 0.004 x 0.008 x 0.0225 P = 0.0182 or 1.82% 2
- 23.