Question 318132: Help me please to resolve these application and extensions
1) You pay $1 to toss 2 coins. If you toss 2 heads, you get $2 (including your first $1); if you toss only 1 head, you get back your $1; and if you toss no heads, you lose your $1. Is this a fair game to play? Why? (Show work please.)
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2) In a raffle 1000 tickets are being sold at $1.00 each. The first prize is $100, and there are 3 second prizes of $50 each. By how much does the price of a ticket exceed its expected value? Why? (Show work please.)
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3) A fair coin is tossed 3 times, and a player wins $3 if 3 tails occur, wins $2 if 2 tails occur, and loses $3 if no tail occur. If 1tail occurs, no one wins. Why? (Show work please.)
a) What is the expected value of the games?
b) Is the game fair?
c) If the answer to part (b) is “No,” how much should the player win or
lose for a toss of exactly 1 tail to make the game fair?
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4) A department store wants to sell 11 purses that cost the store $40 each and 32 purses that cost the store $10 each. If all pursers are wrapped in43 identical boxes and if each customer picks a box randomly find;
a) Each customer’s expectation.
b) The department store’s expected profit if it charges $15 for each box.
(Show work please.)
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5) According to the 2000 U.S. Census, 50.9% of the U.S. population is female. NBC news reported that 1 out of 12 males, but only 1 out of 250 females is color-blind. Given that a person randomly chosen from the U.S. population is color blind, what is the probability that the person is a male? (Show work please.)
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6) In a certain small town, 16% of the population developed lung cancer. If 45% of the population are smokers, and 85% of those developing lung cancer are smokers, what is the probability that a smoker in this population will develop lung cancer. (Show work please.)
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7) A hospital billing department knows that the probability patients 60 or older pay the balance of their bill after one billing is80%, while for a person under the age of 60 the probability is 45%. Seventy percent of the hospital’s patients are 60 years older.
a) what is the probability the balance is paid after one billing? (Show
work please.)
b) The balance of the bill is not paid after one billing. What is the
probability the patients was 60 years or older? (Show work please.)
Thanks to all for your excellent job.
Found 2 solutions by edjones, jrfrunner:
Answer by edjones(8007)   (Show Source):
Answer by jrfrunner(365)  (Show Source):
You can put this solution on YOUR website! 1) You pay $1 to toss 2 coins. If you toss 2 heads, you get $2 (including your first $1); if you toss only 1 head, you get back your $1; and if you toss no heads, you lose your $1. Is this a fair game to play? Why? (Show work please.)
outcome probability net pay
--------- ------------ --------
HH 0.25 1
TH+ HT 0.25+0.25=0.5 0
TT 0.25 -1
Expected payout = 1*0.25+0*0.5+(-1)*0.25 = 0 thus its a fair game
on average the game is not biased in either direction.
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2) In a raffle 1000 tickets are being sold at $1.00 each. The first prize is $100, and there are 3 second prizes of $50 each. By how much does the price of a ticket exceed its expected value? Why? (Show work please.)
outcome probability net pay
--------- ------------ --------
First prize 1/1000 99
2nd price 3/1000 49
Others 996/1000 -1
Expected payout = (1/1000)*99 + (3/1000)*49 +(-1)*996/1000 = -0.75
price of the ticket exceeds the expected value by $0.25
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3) A fair coin is tossed 3 times, and a player wins $3 if 3 tails occur, wins $2 if 2 tails occur, and loses $3 if no tail occur. If 1tail occurs, no one wins. Why? (Show work please.)
a) What is the expected value of the games?
b) Is the game fair?
c) If the answer to part (b) is “No,” how much should the player win or
lose for a toss of exactly 1 tail to make the game fair?
outcome probability Win
--------- ------------ --------
3 tails TTT 3C0*1/8=1/8 3
2 tails TTH 3C1*1/8=3/8 2
1 tail THH 3C2*1/8=3/8 0
0 tsils HHH 3C3*1/8=1/8 -3
a) expected value of game = 3*(1/8)+2*(3/8)+0*3/8+(-3)*1/8 = 6/8=3/4
b) since the expected value of the game is not 0, the game is not fair
c) to make the game fair the expected value =0, therefore 1 tail should be a loss of $2.
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4) A department store wants to sell 11 purses that cost the store $40 each and 32 purses that cost the store $10 each. If all pursers are wrapped in43 identical boxes and if each customer picks a box randomly find;
a) Each customer’s expectation.
b) The department store’s expected profit if it charges $15 for each box.
(Show work please.)
a) customer's expectations? Do you mean the Expected cost?
this would be $40*(11/43)+$10(32/43) =$17.67
b) Profit = $15*43 =$645
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5) According to the 2000 U.S. Census, 50.9% of the U.S. population is female. NBC news reported that 1 out of 12 males, but only 1 out of 250 females is color-blind. Given that a person randomly chosen from the U.S. population is color blind, what is the probability that the person is a male? (Show work please.)
Given: P(F)=0.509, P(C/M)=1/12, P(C/F)=1/250
P(M)=1-.509=.491
Find P(M/C) = P(M and C)/P(C) = P(C/M)*P(M)/(P(C/M)*P(M)+P(C/F)*P(F))
= 1/12*(0.491)/(1/12*(0.491)+1/250*0.509) = 0.953
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6) In a certain small town, 16% of the population developed lung cancer. If 45% of the population are smokers, and 85% of those developing lung cancer are smokers, what is the probability that a smoker in this population will develop lung cancer. (Show work please.)
Given P(L)=0.16, P(S)=0.45, P(S/L)=0.85
Find P(L/S) = P(L and S)/P(S) = P(S/L)*P(L)/P(S) = 0.85*0.16/0.45=0.302
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7) A hospital billing department knows that the probability patients 60 or older pay the balance of their bill after one billing is80%, while for a person under the age of 60 the probability is 45%. Seventy percent of the hospital’s patients are 60 years older.
a) what is the probability the balance is paid after one billing? (Show
work please.)
b) The balance of the bill is not paid after one billing. What is the
probability the patients was 60 years or older? (Show work please.)
let O=patients over 60, U= patients under 60, P=pays after one billing N=does not pay after one billing
Given P(O)=0.70, P(P/O)=0.80, P(P/U)=0.45
therefore P(U)=1-0.70=0.30
a) P(P)=P(P/O)*P(O) + P(P/U)*P(U) = 0.8*0.7 + 0.45*(0.3) = 0.695
and P(N)=1-P(P) = 1-0.695=0.305
b)P(O/N)=P(O and N)/P(N) = P(N/O)*P(O)/P(N) = (1-0.8)*0.70/0.305=0.459
Wheww!!!!!! you need to ask one two questions, I think you are asking to get your homework done which is not the intent in this site
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