and the correct answer is D.

Transcription

1 @. Assume the pobability of a boy being bon is the same as a gil. The pobability that in a family of 5 childen thee o moe childen will be gils is given by A) B) C) D) Solution: The pobability of a gil being bon is 0.5. The pobability of 3 gils and 2 boys being bon is! and the numbe of ways this can happen is C(53). Theefoe the pobability of exactly 3 gils and 2 boys being bon is. The pobability of exactly 4 gils and boy is and " the pobability of exactly 5 gils is. The pobability of 3 o moe gils is the sum: $ &. Since the coect answe is C. 2. A affle is being given by a local chuch. Thee will be one fist pize awaded two second pizes (of $75.00) 5 thid pizes (of $50.00) and 0 fouth pizes (of $25.00). 000 tickets will be sold fo dolla each. How much must the fist pize be in ode fo the affle to be fai? A) 00 B) 200 C) 50 D) 350 Solution: Let ( )* + be the amount of the fist pize. If the game is fai the expected value of winnings is zeo: -. 0/ +$ multiplying both sides by 000 we have o 8 $ $. Theefoe and the coect answe is D. / 7$ Anothe way of aiving at the same answe is to note that in a fai game the sum of the pizes must add up to the ticket evenue. 3. A buye fo a local depatment stoe is consideing buying a batch of clothing fo $ She estimates that the stoe will be able to sell the clothing fo $ with pobability 0.30 $ with pobability 0.50 and $ with pobability If she gets a 0 commission (based on pofit) on the sales of the clothing she can expect to eceive A) B) C) D) Solution: The expected value fo the sales is ( The esulting pofit is 8 coect answe is B. 4. Given?; +& 9$ 523. : 3238 ; 3238 $8 & and?cbd@a E$ 8 and he commission would be 42< =$. The which of the following is coect A) A and B ae independent B) A and B ae mutually exclusive C) A and B ae dependent D) A and B ae complementay?fbg@a H Solution: Since?NKO@A E$?; P2. Since ae independent. The coect answe is A.?; J 86L +$ 92+ $?QKR@A E?; J?&KC@A which? gives which means that and

2 b! { a b 5. E + Coss-multiplying answe is T. and?: E2?UKH@A 5V then?wkx@a E2?; Since?[K\@A OV The (A) Tue (B) False 6. A andom vaiable has mean ^ is geate than _ and standad deviation M A) B). 8 C) / D) Solution: We use the Chebycheff inequality. Fist we find the value of b by setting! _g ^ which means that b dfe. The pobability that Ya & /. 8 8 The coect answe is B.. The pobability that 8 bc is theefoe geate than $ 9\ Ta a 7. A Benoulli distibution with 00 objects and a pobability of success equal to 0.2 can be eplaced with a nomal distibution with A) ^ D) ^ $ $ 8 U Solution: With h o i o 86 j ] B) ^ $ 8 $ ] we have ^ and k. The coect answe is C. 8. Suppose X is a nomally distibuted andom vaiable with ^ and with a standad nomal distibution then Y is the same as A) D) $s[ $s[ B) $s[ Solution: The standad nomal vaiable is given by s The coect answe is A. m hrj q C) ^ C) vuxwzy Y } m - $st uxw d!. $ l Y2m n] $ q U and ]o hrjpk $. Let Z be a andom vaiable $st

3 / 9. If of the eggs in a lage shipment ae boken what is the pobability that a custome checks 3 catons befoe finding a fouth caton with no boken eggs. Solution: The pobability that an egg will be boken is given by 0.0. The pobability that an egg will be unboken is The pobability that a caton of 2 eggs will be unboken is / / d![~ 8. The d!. The pobability pobability that one o moe eggs will be boken in the caton is / / z that one will get 3 catons (with one o boken eggs) and then one unboken caton is 8 Y / /. The answe is Fom solving the bithday poblem in class we know that the pobability of 2 o moe people having the same bithday in a goup of 25 people is appoximately Find the pobability that exactly two people have the same bithday in a goup of 25 people. (Assume 365 days in a yea) Solution: Thee ae ways fo exactly 2 people to have the same bithday and the est diffeent bithdays. Thee ae 365 bithdays fo the 2 people to shae and combinations of bithdays fo the emaining 23 people. The pobability is theefoe 23 : ! $ x24=25 23! $! $. The answe is This is easonable compaed to Given the following table Dives Dives in Goup Stopped fo Moving Violation Goup I (using seat belts) Goup II (not using seat belts) If a dive is stopped fo a moving violation what is the pobability that a) He o she will have a seat belt on? b) He o she will not have a seat belt on? Solution: By looking at the tee diagam below with S being the event of weaing seatbelts and V the event of being ticketed with a moving violation: S V V 0.995

4 The answe to a) is :S ƒa E $ The answe to b) is S ƒ= + 8 E $ Note these must sum to.0! 2. Given the following tee diagam 0.2 A D D c 0.4 B 0.3 D 0.7 D c Find the 0.4 C 0.5 D 0 5 D c. Solution: By a staightfowad application of Bayes E. ". " $. 3. An expeiment consists of thowing two standad 6 sided dice on the table and then taking the poduct of the numbes thown. If is the andom vaiable denoting the poduct of the die The table of the possible values fo is given below: X The fequency distibution is 8 / ˆ ˆ 8 ˆ ˆ ˆ 8 a) Gaph the pobability distibution (histogam) associated with.

5 / Histogam - Fequency Distibution fo the Poduct of Two Dice X b) What is ( ( 9$ 6 8 X 8 $ 4. An un contains 5 ed balls 6 white balls and 4 blue balls. What is the pobability that 2 ed balls and 2 white balls will be dawn in the fist 5 balls. Solution: Note thee ae thee possibilities: 2-ed 2-white -blue; 3-ed 2-white; 2-ed 3-white. Thee ae ways to dawing 5 balls fom 5. The pobability that one gets 2-ed 2-white and -blue is 8 The pobability that one gets 3-ed and 2-white is 8 The pobability that one gets 2-ed and 3-white is 8 The answe is the sum of these pobabilities: 8 8 8