3.3 Probability Distribution(p102)
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1 3.3 Probability Distribution(p102)
2 Mammals: Warm blooded Milk to feed young Take care of young Body hair 3 middle ear bones 3.3 Probability Distribution(p102)
3 3.3 Probability Distribution(p102) Probability Distributions (PD) Discrete PD Binomial PD Continuous PD Uniform PD Normal PD t- PD Chi-square- PD F- PD
4 A certain type of daisy occurs in two variations, one with white flowers and one with red flowers. At the seedling stage it is impossible to determine whether the plant will eventually bear white or red flowers, but it is known that 20% of all seedlings bear red flowers. Suppose you randomly select 12 plants and X is the random variable denoting the number of plants which will bear red flowers. It is estimated that 70% of a potato crop is good, the remainder having rotten centers that cannot be detected unless the potatoes are cut open. 10 potatoes are randomly chosen and tested (cut open). Suppose Y is the number of good potatoes According to an article in the February issue of Readers Digest, South Africans face a 1 in 20 chance of acquiring an infection while hospitalized in the Arthur Blake hospital. The records of 15 randomly selected hospitalized patients are selected randomly and eamined. Let Z be the number of patients that acquired an infection.
5
6 The eperiment consists of a fied number, n, of identical trials. Each trail results in one of two outcomes: success (S) or failure (F). The probability of success on a single trial is equal to some value p and remains the same from trial to trial. The probability of a failure is equal to q = 1-p. The trials are independent. The random variable of interest is the discrete variable, Y, the number of successes observed during the n trials.
7 A random variable X is said to have a binomial distribution based on n trials with success probability p if and only if P( X ) C p (1 p) n n 0,1,2,...,n Where = number of successes p = probability of a success on one trial n = number of trials n C n!!( n )!
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9 The E-Coli Bacteria 3.4 The Binomial Distribution(p103)
10 3.4 The Binomial Distribution: Eample 3.5(p105) Suppose an antibiotic has been shown to be 70% effective against a common bacteria. If the antibiotic is given to 5 unrelated individuals with the bacteria. Let X = the number of individuals for which the antibiotic was effective 2 possible outcomes: effective = success and NOT effective = failure The eperiment consists of a sequence of n = 5 identical trials The probability of a success (effective), denoted by p = 0.7, does not change from trial to trial (is constant/the same for each individual) The trials are independent: unrelated individuals Binimial: n= 5 p = 0.7 P( X ) N C n p (1 p) n N! n!( N p n)! (1 p) n 3.6
11 Suppose an antibiotic has been shown to be 70% effective against a common bacteria. If the antibiotic is given to 5 unrelated individuals with the bacteria, what is the probability that it will be effective in a. eactly three individual? b. at most three individual? c. less than 2 individual? d. more than 1 individual? 3.4 The Binomial Distribution: Eample 3.5(p105) Let X = the number of individuals for which the antibiotic was effective
12 P( X ) N C n p (1 p) n N! n!( N p n)! (1 p) n 3.6
13 P(X is less than 3) = P(X < 3) =P(X 2 ) P(X is at least 3) = P(X 3) = 1- P(X 2 ) P(X more than 3) = P(X > 3) = 1- P(X 3 ) P(X at most 3) = P(X 3)
14 Let X = the number of individuals for which the antibiotic was effective what is the probability that it will be effective in a. eactly three individual?
15 Let X = the number of individuals for which the antibiotic was effective what is the probability that it will be effective in b. at most three individual? P ( X 0 or X 1or X 2 or X 3) P[ X 0 X 1 X 2 X 3) P( X 0) P( X 1) P( X 2) P( X 3)
16 Let X = the number of individuals for which the antibiotic was effective what is the probability that it will be effective in c. less than 2 individual? d. more than 1 individual?
17 Mean( X ) np Var( X ) 2 np(1 p) Mean number of successes can also read as the epected number of successes.
18 3.4 The Binomial Distribution: Properties A multiple choice test has four possible answers to each of 16 questions. A student guesses the answer to each question. X is the number of correct answers. A bo contains 5 defective and 10 non-defective globes. A sample of 3 globes are tested. Let Z be the number of defective globes. A game in which a true die is thrown and if a 4, 5, or 6 comes up you win; on a 1, 2, or 3 you lose. Betty is playing the game until she has 3 wins, the one after the other. X is the number of wins. The probability that you will not get flu when injected against it at the beginning of the winter is 0,75. Twenty randomly chosen people are injected at the start of the winter. Five of these people also drink on a daily bases vitamin C tablets. X is the number of people that do not get flu during that winter.
19 3.4 The Binomial Distribution: Problem 19(p143) n p
20 3.4 The Binomial Distribution: Class eecise It is known that 40 percent of the mice used in an eperiment will become aggressive within one minute after having been administered an eperimental drug. Ten mice have been administered the drug. Find the probability that: a. Eactly four become very aggressive within one minute. b. At most four become very aggressive within one minute. c. At least 4 become very aggressive within one minute. d. What is the epected number of mice that will become very aggressive within one minute?
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