Question

12. Zac raises funds for a library by running a game where players spin a needle. The final position of the needle results in an outcome where a player wins or loses money. The outcomes, with associated probabilities, are shown in the following diagram. diagram not to scale Win $5 40% chance Lose $8 10% chance Lose $5 20% chance Lose $10 30% chance Let X represent the amount that a player of this game wins. (a) (i) Find the expected value of X. (ii) Interpret your answer to part (a)(i). To encourage a person to keep playing this game, Zac increases the winning prize for the second game they play from $5 to $6. For each successive game they play, the winning prize continues to increase by $1. Emily plays k games. The kth game is fair. (b) (i) Find the value of k. (ii) Explain why Zac expects to raise money from the games Emily plays.

          12.
Zac raises funds for a library by running a game where players spin a needle. The final position of the needle results in an outcome where a player wins or loses money. The outcomes, with associated probabilities, are shown in the following diagram.
diagram not to scale

Win $5
40% chance

Lose $8
10% chance

Lose $5
20% chance

Lose $10
30% chance

Let X represent the amount that a player of this game wins.
(a) (i) Find the expected value of X.
(ii) Interpret your answer to part (a)(i).

To encourage a person to keep playing this game, Zac increases the winning prize for the second game they play from $5 to $6. For each successive game they play, the winning prize continues to increase by $1.

Emily plays k games. The kth game is fair.
(b) (i) Find the value of k.
(ii) Explain why Zac expects to raise money from the games Emily plays.

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12.
Zac raises funds for a library by running a game where players spin a needle. The final position of the needle results in an outcome where a player wins or loses money. The outcomes, with associated probabilities, are shown in the following diagram.
diagram not to scale

Win 5
40% chance

Lose8
10% chance

Lose 5
20% chance

Lose10
30% chance

Let X represent the amount that a player of this game wins.
(a) (i) Find the expected value of X.
(ii) Interpret your answer to part (a)(i).

To encourage a person to keep playing this game, Zac increases the winning prize for the second game they play from 5 to6. For each successive game they play, the winning prize continues to increase by 1.

Emily plays k games. The kth game is fair.
(b) (i) Find the value of k.
(ii) Explain why Zac expects to raise money from the games Emily plays.

Added by Dasti U.

STATS Modeling The World

David E. Bock, Paul F. Velleman, Richard… 4th Edition

Chapter 5

Instant Answer

Solved by Expert Daniel Carr

08/28/2024

Step 1

- Win $5 with a probability of 40% (0.4) - Lose $5 with a probability of 20% (0.2) - Lose $8 with a probability of 10% (0.1) - Lose $10 with a probability of 30% (0.3) Show more…

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Transcript

00:01 So for part a, step one here, on the expected value of x, if x is the represent the amount of player wins in the game, our expected value of x is going to be each probability multiplied by its winnings, so our losings in most cases.

00:16 So we've got a 40 % chance.

00:18 That's a 0 .4 times 5 plus 0 .1 times a negative 8 plus 0 .2 times negative 5.

00:31 Plus 0 .3 times a negative 10.

00:34 So our expected value is going to be 0 .4 times 5, plus 0 .1 times a negative 8, plus 0 .2 times a negative 5, plus 0 .3 times a negative 10.

00:46 And that comes out to a negative 2 .8.

00:49 So the idea of the interpretation here is that for every...

00:59 Let's here.

01:00 The final position...

01:05 Okay, so for every, for every time playing, you can expect to lose $2 .80.

01:24 Anytime you play, that's the expected case there.

01:27 Okay, so for part b, if k represents a number of games until the game is fair, meaning our expected return is going to be zero.

01:36 Well, that's going to be, let's see here.

01:40 So if we look at it as it would be 0 .4 times 6.

01:47 Sorry, it's actually going to be this.

01:48 It's going to be 0 times.

01:50 We're going to put a k here, plus 0 .1 times a negative 8, plus 0 .2 times a negative 5, plus 0 .3 times a negative 10.

01:59 So we'll be able to solve for k through this.

02:01 So 0 is equal to 0 .4k, plus we've got 0 .1 times a negative 8, plus 0 .2.

02:06 2 times a negative 5 plus 0 .3 times a negative 10.

02:10 It comes out to a negative 4 .8.

02:13 We add that to the other side, so 4 .8 equals 0 .4k.

02:18 I'm going to divide all sides by 0 .4...

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