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# Intro To Prob & Statistics MATH 2311

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This 4 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 2311 at University of Houston taught by Shahinda Hafeez in Fall. Since its upload, it has received 42 views. For similar materials see /class/208388/math-2311-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

Wm 1 W The Binomial Distribution A binomial experiment has the following properties 1 Number of trials is xed 2 There are 2 outcomes of the experiment Success probability denoted by p and failure probability denoted by q 3 The probability of success in each trial is the same 4 The trials are independent of each other Experiments with two outcomes are called Bernoulli trials or Binomial trials Finding the Probability of an Event of a Binomial Experiment In a binomial experiment in which the probability of sucCess in any trial is p the probability of exactly x successes in n independent trials is given by quotif 39 PXxCnxpquotqquotquot X is called a binomial random variable and its probability distribution is called a 39 binomial probability distribution Example 1 on page 505 derives this formula Example 1 Consider the following binomial experiment A fair die is cast four times Compute the probability of obtaining exactly one 6 in the four throws 631 Wham at ammo to ma Cfquotth 3 iii93 in is cm ca can a Camelquot quoti 133 23 Math 1313 Section 84 2 Example 2 Let the random variable X denote the number of girls in a vechild family If the probability of a female birth is 05 construct the binomial distribution associated with this experiment Tn 39312 if I x 7 S l i r l 3 w 53 2 i s f w 22 w mmquot a 39 ha WM 06me man a lawmilks l 5 3 if it P 12 ch123 C Siz 932 is Z v37l Wm craze 65gt lt51 W grqugtlr l p QC X 0 03 of is MC 3 P 5 i O i PO M g ml Cw 2 CS lt 100 What is the probability of having at least one girl FYXT B l RX 2 39lquot if J i or I 190640 a 03 9amp0 VMean Variance and Standard Deviation of a Random Variable If X is a binomial random variable associated with a binomial experiment consisting of 21 trials with probability of success p and probability of failure q then the mean EX variance and stande deviation of X are given by applying the following formulas Exam1Ple 2 M 5653 25 may Bosnia L25 WW4me crJVarxc g 3 U ZEE 4 HEX P Varx npq Math 1313 Section 84 3 Example 3 Consider the following binomial experiment If the probability that a marriage will end in divorce within 20 years after its start is 06 what is the probability that out of 6 couples just married in the next 20 years all will be divorced 219 car lt4 n to 21 None will be divorced szo 2 Clue0 070 MY vooktt b Exactly two couples will be divorced 3922 use to f LkiY 2332 C At least two couples will be divorced POtzz B39 139quot 9CX gt lt U eEoooe 2 a 33me cup 3 mommy o3ea 2 56 590 g 4 er I 39 d At most ve couples Will be divorce w x m Duh7 HMS 2 39 690057 iv FCXcol la a v W Vii 690a ear atteon m we 0 gm lq S 3 V Example 4r A baseball pitChef gives up a hit on the average of once every 39 hpitch If nine pitches are thrown what is the probability that l 5quot 1301 mgt Ohms a Exactly three of them result in a hits m ab 539 39 ijipz PCX B k CCCM C S 2 b None of them results in a hit 0 Cl P6010 CCUWB 23 83 Pl c No more than seven result in hits l N I M 1 PM 373 HDOQ 8 la eet 5305ij C e M Q QC PCX e7 Cemce wga 4000002 39Oquot Mar cm lt1 a we if 300000 c variance p 0L 2 i2a Math 1313 Section 84 4 Example 5 A machine produces light bulbs to meet certain speci cations and 80 of the light bulbs meet these speci cations A sample of six bulbs is taken from the machine 3 production and placed in a box What is the probability that three or more of them fail to meet speci cations W w quot 2 4 2 v o v to Q lt1 i I r g m Pop a quott Ma Q PO 139 n quotN quot quot 323 3 17 1 Id 5 f i if M M Mqu wi W ow p w Example 6 The probability of a person contracting in uenza on exposure is 62 In the binomial experiment for a family of 12 that has been exposed what is the a mean h ho I1 1265923 744 MW WW39WWM 2 W2 39 0 g Li b standarddeviation WV lt C9Z303Xj r N N W383 2862272 9 V

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