ASSIGNMENT 3 1 Statistical Methods Probability M S 124 125 177 134 137 NAME 3 9 3 points An experiment result in one of the following sample points E1 E2 E3 E4 and E5 a Find P E3 if P E1 1 P E2 2 P E4 1 and P E5 1 b Find P E3 if P E1 P E3 P E2 1 P E4 2 and P E5 1 c Find P E3 if P E1 P E2 P E4 P E5 1 ANSWERS a 5 b 3 c 6 3 15 5 points Two marbles are drawn at random and without replacement from a box containing two blue marbled and three red marbles a List the sample points b Assign probabilities to the sample points c Determine the probability of observing each of the following events A Two blue marbles are drawn B A red and a blue marble are drawn C Two red marbles are drawn ANSWERS a R1R2 R1R3 R2R3 R1B1 R1B2 R2B1 R2B2 R3B1 R3B2 B1B2 b 1 10 for each sample point c P A 1 10 P B 6 10 P C 3 10 3 19 2 points USDA chicken inspection The United States Department of Agriculture USDA reports that under its standard inspection system one in every 100 slaughtered chickens passes inspection for fecal contamination Tampa Tribune Mar 31 2000 a If a slaughtered chicken is selected at random what is the probability that it passes inspection for fecal contamination b The probability of part a was based on a USDA study which found that 306 of 32 075 chicken carcasses passed inspection for fecal contamination Do you agree with the USDA s statement about the likelihood of a slaughtered chicken passing inspection for fecal contamination ANSWERS a 01 b yes 3 24 4 points Sex composition patterns of children in families In having children is there a genetic factor that causes some families to favor one sex over the other That is does having boys or girls run in the family This was the question of interest in Chance Fall 2001 Using data collected on children s sex for over 4 000 American families that had at least two children the researchers complied the accompanying table An American family with at least two children is selected and the sex composition of the first two children is observed a List the sample points for this experiment Sex Composition of First Two Children Frequency Boy Boy Boy Girl Girl Boy Girl Girl 1 085 1 086 1 111 926 TOTAL 4 208 Source Rodgers J L and Doughty D Does having boys or girls run in the family Chance Vol 14 No 4 Fall 2001 Table 3 b If having boy is no more likely than having a girl and vice versa assign a probability to each sample point c Use the information in the table to estimate the sample point probabilities Do these estimates agree to a reasonable degree of approximation with the probabilities you found in part b d Make an inference about whether having boys or girls runs in the family ANSWERS 3 181 2 points Odd Man Out Three people play a game called Odd Man Out In this game each player flips a fair coin until the outcome heads or tails for one of the players is not the same as that for the other two players This player is then the odd man out and loses the game Find the probability that the game ends i e either exactly one of the coins will fall heads or exactly one of the coins will fall tails after only one toss by each player Suppose one of the players hoping to reduce his chances of being the odd man out uses a two headed coin Will this ploy be successful Solve by listing the sample points in the sample space ANSWER Probabilities are the same 3 44 7 points Consider the following Venn diagram where P E1 10 P E2 05 P E3 P E4 2 P E5 06 P E6 3 P E7 06 and P E8 03 E1 E3 A E2 E4 B E7 E5 E8 E 6 E6 B Find the following probabilities a P Ac b P Bc c P Ac B d P A B e P A B f P Ac Bc g Are event A and B mutually exclusive Why ANSWERS S 3 45 8 points The outcomes of two variables are Low Medium High and On Off respectively An experiment is conducted in which the outcomes of each of the two variables are observed The probabilities associated with each of the six possible outcome pairs are given in the following table On Off Low Medium High 50 25 10 07 05 03 Consider the following events A On B Medium or On C Off and Low D High a Find P A b Find P B c Find P C d Find P D e Find P Ac f Find P A B g Find P A B h Consider each possible pair of events taken from the events A B C and D List the pairs of event that are mutually exclusive Justify your choices ANSWERS a 65 b 72 c 25 d 08 e 35 f 72 g 65 h A and C B and C C and D 3 50 4 points Binge alcohol drinking A study of binge alcohol drinking by college students was published in the American Journal of Public Health July 1995 Suppose an experiment consists of randomly selecting one of the undergraduate students who participated in the study Consider the following events A The student is a binge drinker B The student is a male C The student lives in a coed dorm Describe each of the following events in terms of unions intersections and complements A B A B Ac etc a The student is male and a binge drinker b The student is not a binge drinker c The student is male or lives in a coed dorm d The student is female and not a binge drinker ANSWERS 3 56 7 points Gang research study The National Gang Crime Research Center NGCRC has developed a six level gang classification system for both adults and juveniles The NGCRC collected data on approximately 7 500 confined offenders and assigned each a score using the gang classification system Journal of Gang Research Winter1997 One of several other variables measured by the NGCRC was whether or not the offender ever carried a homemade weapon e g a knife while in custody The table on p 138 gives the number of confined offenders in each of the gang score and homemade weapon categories Assume one of the confined offenders is randomly selected a Find the probability that the offender has a gang score of 5 b Find the probability that the offender has carried a homemade weapon c Find the probability that the offender has a gang score below 3 d Find the probability that the offender has a gang score of 5 and has carried a homemade weapon e Find the probability that the …