Further Statistics 1 - Discrete Probability Distributions HELP

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confuzzledteen
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Caitlin is designing a game of chance for her school fete.
She has a fair, five-sided spinner marked with the numbers 1, 2, 3, 4 and 5. Players get 20 virtual points to have a go at spinning an even number.
If they are successful, they win their points back plus k times the number spun. Points won can then be exchanged for small prizes.
Given that Caitlin’s expected winnings per game is 3 points, show that k = 7.5.

I can't seem to work out how to answer this, any help would be appreciated! Currently revising for a test..
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Theloniouss
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What have you tried?
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confuzzledteen
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It's a 4 marker,,

so I've gotten as far as assuming that 1 mark comes from writing the probability distribution out:
x 1 2 3 4 5
P(X=x) 0.2 0.2 0.2 0.2 0.2

Would it be appropriate to use the binomial distribution X~(20,0.4) where n=20 & p=0.4?
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Theloniouss
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The probability of getting an even number is fixed and the spinner is being spun until they get an even number, which means the distribution is probably geometric.
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confuzzledteen
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Oh alright, that's true.. How would I model this with an equation?
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Theloniouss
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(Original post by confuzzledteen)
Oh alright, that's true.. How would I model this with an equation?
Can you take a picture of the question? I don't think I've quite understood it, actually.
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confuzzledteen
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Hope you can access that..
In this chapter, we learn about Expected Value whatnot so maybe that could be used?
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DFranklin
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(Original post by confuzzledteen)
It's a 4 marker,,

so I've gotten as far as assuming that 1 mark comes from writing the probability distribution out:
x 1 2 3 4 5
P(X=x) 0.2 0.2 0.2 0.2 0.2

Would it be appropriate to use the binomial distribution X~(20,0.4) where n=20 & p=0.4?
It's not binomial or geometric. For each value of x, how much does she win? (this will be a function of k and x).
Then the expected winnings are \sum_{x=1}^5 P(X=x)W(x), where for each x, W(x) is the amount she wins for throwing that value.
Since you now know the expected winnings, you can solve for k.
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Theloniouss
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I'm really not sure? I calculate the player's expected points to be 17, and Caitlin's expected points to be 12, which must be wrong.
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confuzzledteen
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(Original post by DFranklin)
It's not binomial or geometric. For each value of x, how much does she win? (this will be a function of k and x).
Then the expected winnings are \sum_{x=1}^5 P(X=x)W(x), where for each x, W(x) is the amount she wins for throwing that value.
Since you now know the expected winnings, you can solve for k.
I like your method - it does make sense to me.. But I still can't seem to get k=7.5; could you tell me where I'm going wrong? Like I know what I have to do, I think I'm just doing the starting bit wrongly..

For each value of x, how much does she win? I said that for x=1, she wins back 20+k and for x=2, she wins back 20+2k and for x=3, she wins back 20+3k etc. but this doesn't seem to give me the right answer.. I think I'm processing the data given wrongly
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confuzzledteen
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(Original post by Theloniouss)
I'm really not sure? I calculate the player's expected points to be 17, and Caitlin's expected points to be 12, which must be wrong.
Are you going about it via @DFranklin 's method? It seems right to be (as it ties into what the chapter is about, i.e. expected value) but I just can't seem to write down the correct equation to start me off
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Theloniouss
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(Original post by confuzzledteen)
Are you going about it via @DFranklin 's method? It seems right to be (as it ties into what the chapter is about, i.e. expected value) but I just can't seem to write down the correct equation to start me off
Yep, I got -5 via that method.
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confuzzledteen
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(Original post by Theloniouss)
Yep, I got -5 via that method.
It feels as if it's the right method tho.. Maybe our function of x and k for each value of x is wrong?
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DFranklin
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It says you only win if you spin an even number.

(It's actually really unclear whether you "bet 20 points", or "bet 1 point" (with 20 being irrelevant to this part of the question) which will affect how much you win / lose).
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Theloniouss
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(Original post by confuzzledteen)
It feels as if it's the right method tho.. Maybe our function of x and k for each value of x is wrong?
(Original post by DFranklin)
It says you only win if you spin an even number.

(It's actually really unclear whether you "bet 20 points", or "bet 1 point" (with 20 being irrelevant to this part of the question) which will affect how much you win / lose).
I suspect there's a problem with the question. No method I can think of makes k=7.5
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confuzzledteen
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(Original post by DFranklin)
It says you only win if you spin an even number.

(It's actually really unclear whether you "bet 20 points", or "bet 1 point" (with 20 being irrelevant to this part of the question) which will affect how much you win / lose).
You're right there.. dumb question
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DFranklin
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OK, I can get 7.5 using my method.

Clarified rules:
People pay 20 units to play.
If they throw odd, they lose.
Otherwise they get the 20 back, plus the k times the "number rolled".

Finally, the winnings of 3 are for Caitlin. (In other words, that means the person playing has an expected loss of 3).
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confuzzledteen
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(Original post by DFranklin)
OK, I can get 7.5 using my method.

Clarified rules:
People pay 20 units to play.
If they throw odd, they lose.
Otherwise they get the 20 back, plus the k times the "number rolled".

Finally, the winnings of 3 are for Caitlin. (In other words, that means the person playing has an expected loss of 3).
Oh that's grand! You're a legend. Do you mind showing your working/calc below?
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DFranklin
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(Original post by confuzzledteen)
Oh that's grand! You're a legend. Do you mind showing your working/calc below?
I'm not going to do the question for you (which is the forum rule). You should have enough to do it yourself. Post your working if you get stuck.
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Muttley79
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(Original post by confuzzledteen)
I like your method - it does make sense to me.. But I still can't seem to get k=7.5; could you tell me where I'm going wrong? Like I know what I have to do, I think I'm just doing the starting bit wrongly..

For each value of x, how much does she win? I said that for x=1, she wins back 20+k and for x=2, she wins back 20+2k and for x=3, she wins back 20+3k etc. but this doesn't seem to give me the right answer.. I think I'm processing the data given wrongly
Surely you win nothing for a 1, 3 or 5 - that's how I read it
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