Joe buys a ticket in the Tri-State Pick 3 lottery every day, always betting on 9

Question

Joe buys a ticket in the Tri-State Pick 3 lottery every day, always betting on 956. He will win something if the winning number contains 9, 5, and 6 in any order. Each day, Joe has probability 0.007 of winning, and he wins (or not) independently of other days because a new drawing is held each day. What is the probability that Joe's first winning ticket comes on the 29th day? Sheila's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels du ring pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per (mg/dl) one hour after a sugary drink is ingested. Sheila's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with mu = 122 mg/dl and sigma = 10 mg/dl, What is the level L such that there is probability only 0.05 that the mean glucose level of 3 test results falls above L for Sheila's glucose level distribution? mg/dl It is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as Benford's Law, shown below. Fake records usually have fewer first digits 1, 2, and 3. What is the approximate probability, if Benford's Law holds, that among 1186 randomly chosen invoices there are no more than 681 in amounts with first digit 1, 2, or 3? The weight of the eggs produced by a certain breed of hen is normally distributed with mean 64.1 grams (g) and standard deviation 5.1 g. If carton of such eggs can be considered to be SRSs of size 12 from the population of all eggs, what is the probability that the weight of a carton falls between 775 g and 825 g?

Explanation / Answer

5) probabilty =loses in first 28 days *wins on 29th day=(1-0.007)28*0.007=0.0058

6)for sample size 3; std error of mean =std deviation/(n)1/2 =5.7735

at 95th percentile z=1.96

hence level L =mean +z*std deviation =133.3

7)probabilty of first digit as 1,2,3 =0.299+0.165+0.116=0.58

hence mean =np=1186*0.58=687.88

std deviation=(np(1-p))1/2 =16.997

hence P(X<=681)=P(Z<(681.5-687.88)/16.997)=P(Z<-0.3754)=0.3537

8)

mean weight of carton =64.1*12 =769.2

std deviation =5.1*(12)1/2 =17.6669

hence P(775<X<825)=P((775-769.2)/17.6669<Z<(825-769.2)/17.6669)=P(0.3283<Z<3.1584)=0.9992-0.6287=0.3706

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