Thanks! Stats (4th Edition) Chapter 15, Problem 14E Show all ste Problem Carniva

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Thanks!

Stats (4th Edition) Chapter 15, Problem 14E Show all ste Problem Carnival A carnival game offers a S100 cash prize for anyone who can break a balloon by throwing a dart at it. It costs $5 to play, and you're willing to spend up to $20 trying to win. You estimate that you have about a 10% chance of hitting the balloon on any throw a) Create a probability model for this carnival game b) Find the expected number of darts you'll throw. c) Find your expected winnings. Step-by-step solution Step 1 of 3 a) Amount offered by Carnival game to break one balloon by throwing a dart at it is $100 There is 10% chances to hit a balloon The probability model for the game is Throws 12 3 4 P (hitting)| 1/10 1/101 1/101 1/10 Comment Step 2 of 3 b) Expected Number of darts you will throw is Comment Step 3 of 3 Expected winnings = 1 (100) = 100$ Comment

Explanation / Answer

Cash Prize = $ 100

Cost of a throw = $ 5

(a) Here we know that if we hit the ballon we will not throw again.We will at max throw 4 dart

so Pr(Hitting baloon in first chance) = 0.10

Pr(Hitting ballon in secon chance) = Pr(failed in first chance) * Pr(Hit in second chance) = 0.90 * 0.10 = 0.9

Pr(Hitting in third chance) = 0.92 * 0.10 = 0.081

Pr(Hitting in fourth chance) = 0.93 * 0.10 = 0.0729

so Pr(I will hit the ballon in 4 chance) = 0.10 + 0.09 + 0.081 + 0.0729 = 0.3439

so Pr(I will be unsuccessful in 4 chance) = 1 - 0.3439 = 0.6561

Here we take random variable X (which is the number of darts throwed)

so Here PMF of X is

P(1) = 0.1 ; X = 1

P(2) = 0.09; X= 2

P(3) = 0.081 ; X = 3

P(4) = 1 - (0.1 + 0.09 + 0.081) = 0.729 ; X = 4

here we take all probability for X = 4 because if we will fail in 1st, 2nd and 3rd attempt , we will definitely try the 4th attempt.

(b) Expected number of throws = 1 * 0.10 + 2 * 0.09 + 3 * 0.081 + 4 * 0.729 = 3,439

(c) Expected winning

When for X =1 , expected winning = 100 - 5 = 95

for X = 2 , expected winning = 100 - 5 * 2 = 90

For X = 3; expected winning = 100 - 5 * 3 = 85

for X = 4; expected winning = 100 - 5 * 4 = 80

Expected earning = 95 * 0.10 + 90 * 0.09 + 85 * 0.081 + 80 * (0.729 * 0.1) - 20 * (0.729 * 0.9) = $ 17.195

[ Here we Pr(X = 4) include two type of probability one is success (0.729 * 0.1) in that attempt in second faiure (0.729 * 0.9) in tat attept

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