. An urn contains red, green, and blue balls. Let r, g, b be the proportions of red, green,

blue balls, respectively (r + g + b = 1).

(a) Balls are drawn randomly with replacement. Find the probability that the first time

a green ball is drawn is before the first time a blue ball is drawn.

Hint: Explain how this relates to finding the probability that a draw is green, given that

it is either green or blue.

(b) Balls are drawn randomly without replacement. Find the probability that the first

time a green ball is drawn is before the first time a blue ball is drawn. Is the answer the

same or di?erent than the answer in (a)?

Hint: Imagine the balls all lined up, in the order in which they will be drawn. Note that

where the red balls are standing in this line is irrelevant.

(c) Generalize the result from (a) to the following setting. Independent trials are performed, and the outcome of each trial is classified as being exactly one of type 1, type 2,

. . . , or type n, with probabilities p1, p2,...,pn, respectively. Find the probability that

the first trial to result in type i comes before the first trial to result in type j, for i 6= j.

A gambler repeatedly plays a game where in each round, he wins a dollar with probability

1/3 and loses a dollar with probability 2/3. His strategy is "quit when he is ahead by

$2," though some suspect he is a gambling addict anyway. Suppose that he starts with

a million dollars. Show that the probability that he'll ever be ahead by $2 is less than

1/4.

46. As in the gambler's ruin problem, two gamblers, A and B, make a series of bets, until

one of the gamblers goes bankrupt. Let A start out with i dollars and B start out with

N

Hint(s) | Check My Work Consider the following hypothesis test: Ho: us 27 Ha: H > 27 A sample of 50 provided a sample mean of 28.4. The population standard deviation is 6. a. Compute the value of the test statistic (to 2 decimals). b. What is the p-value (to 4 decimals)? Use the value of the test statistic rounded to 2 decimal places in your calculations.1. A study was conducted among 96 adults aged 80 years or above living in Hanoi, Vietnam to examine the association between smoking and myocardial infarction (MI). Data are provided in the table below: MI No MI Nonsmokers 10 30 Smokers 8 48 a. State the key scientific question of the study and translate it into a statistical question. b. Conduct the chi-squared test at the 0.05 significance level. Report the test statistic and p-value. You can use Stata for the calculation. c. What do you conclude for the statistical and scientific question, respectively, based on results in (b)? d. What is the risk of having MI among smokers and non-smokers, respectively? e. What is the relative rick (RR) of having MI among smokers than non-smokers? How to quantify the uncertainty of this point estimate? You can use Stata for the calculation. f. What do you conclude for the statistical and scientific question, respectively, based on results in (e)? g. Suppose the scientists who conducted this study secured additional funding to recruit more participants and they found each cell size of the 2 by 2 table was increased by 10-fold. Conduct the chi-squared test at the 0.05 significance level. Report the test statistic and p-value. You can use Stata for the calculation. h. What do you conclude for the statistical and scientific question, respectively, based on results in (g)? i. Report the point and interval estimates of the RR based on information given in (9). You can use Stata for the calculation. j. What do you conclude for the statistical and scientific question, respectively, based on results in ()? k. Compare the results in (e) and (), what do you notice?(4 points] Note: For this Question, It may be necessary to use R to avoid LONG calculations and achieve the required precisions in both answers. For the following pairs of {my} observations. [112)1(3 4)! (t 4}: (gs 3'): [1219}: we are interesting in tting the population regression model \"y = o + 313. Carry out the hypothesis test Ho i31=\":I H1 51%\" Detennine the value (3f thE_tESt statistic and the associated p-ualue. Test Statistic = p-Value = {4 points) Note: Forthis Question, it may be necessary to use R to avoid LUNG calculations and achieve the required precisicns in both answers. Also notice that the slope under the null hypothesis is NOT zero {0}. For the data set we have the following (my) pairs of observations: (21_1}1{39011{Er 4],{918),[QJI]}, Piease SEW out the hypothesis test: Ho 131:1 H1 31%]- Determine the value of the test statistic and the associated pvalue. Test Statistic = ptfalue =