Mixing
You toss two coins. This action costs $5$. For each head tossed, you get $2$. If you win (at least one head...
$begingroup$
You toss two coins. This action costs $ $ 5$. For each head tossed, you get $$2$. If you win (at least one head tossed) you also get your $$5$ back. What is the expected profit of a round? What should be the sum of the award of winning so that the game is fair (no trend of someone winning or losing)?
At this point I have calculated the possible scenarios:
- 1/4 - two heads
- 1/4 - two tails
- 1/2 - one head, one tail
Hence, you have
$$
left(frac{1}{4} text{chance of winning} ($5+$2+$2- $5)right) times left( frac{1}{2} text{chance of winning} ($5+$2- $5) right) times left( frac{1}{4} text{chance of winning} ($0 - $5) right)$$-.
probability statistics
asked Jan 23 at 14:20
Radoslav BonevRadoslav Bonev
414
$endgroup$
add a comment |
$begingroup$
You toss two coins. This action costs $ $ 5$. For each head tossed, you get $$2$. If you win (at least one head tossed) you also get your $$5$ back. What is the expected profit of a round? What should be the sum of the award of winning so that the game is fair (no trend of someone winning or losing)?
At this point I have calculated the possible scenarios:
- 1/4 - two heads
- 1/4 - two tails
- 1/2 - one head, one tail
Hence, you have
$$
left(frac{1}{4} text{chance of winning} ($5+$2+$2- $5)right) times left( frac{1}{2} text{chance of winning} ($5+$2- $5) right) times left( frac{1}{4} text{chance of winning} ($0 - $5) right)$$-.
probability statistics
asked Jan 23 at 14:20
Radoslav BonevRadoslav Bonev
414
$endgroup$
$begingroup$
You get $2$ what?
$endgroup$
– N. F. Taussig
Jan 23 at 14:22
$begingroup$
Typesetting hint: to display dollar signs, put a backslash in front:$5gives $5. Otherwise, everything between your dollar signs is interpreted as a mathematical expression.
$endgroup$
– Théophile
Jan 23 at 14:23
1
$begingroup$
I think you mean $+$ instead of $*$ in your last sentence. Otherwise, you have exactly written out how to calculate the expected profit of the game. It's unclear what you mean by "the sum of the award of winning" in you second question though.
$endgroup$
– kccu
Jan 23 at 14:33
2
$begingroup$
You should have addition signs rather than multiplication signs between each pair of parentheses since the three events are mutually exclusive.
$endgroup$
– N. F. Taussig
Jan 23 at 14:40
1
$begingroup$
This is what happens when you don't take the trouble to ensure that your question is reasonably formatted: someone comes along and formats it for you, and while doing so they change a $+$ to $times$ and make it slightly worse (in that respect) than it was before. I recommend you go over the edited post and make sure everything there is exactly how you want it--notably those multiplication signs.
$endgroup$
– David K
Jan 23 at 14:40
add a comment |
$begingroup$
You toss two coins. This action costs $ $ 5$. For each head tossed, you get $$2$. If you win (at least one head tossed) you also get your $$5$ back. What is the expected profit of a round? What should be the sum of the award of winning so that the game is fair (no trend of someone winning or losing)?
At this point I have calculated the possible scenarios:
- 1/4 - two heads
- 1/4 - two tails
- 1/2 - one head, one tail
Hence, you have
$$
left(frac{1}{4} text{chance of winning} ($5+$2+$2- $5)right) times left( frac{1}{2} text{chance of winning} ($5+$2- $5) right) times left( frac{1}{4} text{chance of winning} ($0 - $5) right)$$-.
probability statistics
asked Jan 23 at 14:20
Radoslav BonevRadoslav Bonev
414
$endgroup$
You toss two coins. This action costs $ $ 5$. For each head tossed, you get $$2$. If you win (at least one head tossed) you also get your $$5$ back. What is the expected profit of a round? What should be the sum of the award of winning so that the game is fair (no trend of someone winning or losing)?
At this point I have calculated the possible scenarios:
- 1/4 - two heads
- 1/4 - two tails
- 1/2 - one head, one tail
Hence, you have
$$
left(frac{1}{4} text{chance of winning} ($5+$2+$2- $5)right) times left( frac{1}{2} text{chance of winning} ($5+$2- $5) right) times left( frac{1}{4} text{chance of winning} ($0 - $5) right)$$-.
probability statistics
probability statistics
asked Jan 23 at 14:20
Radoslav BonevRadoslav Bonev
414
asked Jan 23 at 14:20
Radoslav BonevRadoslav Bonev
414
edited Jan 23 at 14:38
Radoslav Bonev
asked Jan 23 at 14:20
Radoslav BonevRadoslav Bonev
414
asked Jan 23 at 14:20
Radoslav BonevRadoslav Bonev
414
asked Jan 23 at 14:20
Radoslav BonevRadoslav Bonev
414
414
$begingroup$
You get $2$ what?
$endgroup$
– N. F. Taussig
Jan 23 at 14:22
$begingroup$
Typesetting hint: to display dollar signs, put a backslash in front:$5gives $5. Otherwise, everything between your dollar signs is interpreted as a mathematical expression.
$endgroup$
– Théophile
Jan 23 at 14:23
1
$begingroup$
I think you mean $+$ instead of $*$ in your last sentence. Otherwise, you have exactly written out how to calculate the expected profit of the game. It's unclear what you mean by "the sum of the award of winning" in you second question though.
$endgroup$
– kccu
Jan 23 at 14:33
2
$begingroup$
You should have addition signs rather than multiplication signs between each pair of parentheses since the three events are mutually exclusive.
$endgroup$
– N. F. Taussig
Jan 23 at 14:40
1
$begingroup$
This is what happens when you don't take the trouble to ensure that your question is reasonably formatted: someone comes along and formats it for you, and while doing so they change a $+$ to $times$ and make it slightly worse (in that respect) than it was before. I recommend you go over the edited post and make sure everything there is exactly how you want it--notably those multiplication signs.
$endgroup$
– David K
Jan 23 at 14:40
add a comment |
$begingroup$
You get $2$ what?
$endgroup$
– N. F. Taussig
Jan 23 at 14:22
$begingroup$
Typesetting hint: to display dollar signs, put a backslash in front:$5gives $5. Otherwise, everything between your dollar signs is interpreted as a mathematical expression.
$endgroup$
– Théophile
Jan 23 at 14:23
1
$begingroup$
I think you mean $+$ instead of $*$ in your last sentence. Otherwise, you have exactly written out how to calculate the expected profit of the game. It's unclear what you mean by "the sum of the award of winning" in you second question though.
$endgroup$
– kccu
Jan 23 at 14:33
2
$begingroup$
You should have addition signs rather than multiplication signs between each pair of parentheses since the three events are mutually exclusive.
$endgroup$
– N. F. Taussig
Jan 23 at 14:40
1
$begingroup$
This is what happens when you don't take the trouble to ensure that your question is reasonably formatted: someone comes along and formats it for you, and while doing so they change a $+$ to $times$ and make it slightly worse (in that respect) than it was before. I recommend you go over the edited post and make sure everything there is exactly how you want it--notably those multiplication signs.
$endgroup$
– David K
Jan 23 at 14:40
$begingroup$
You get $2$ what?
$endgroup$
– N. F. Taussig
Jan 23 at 14:22
$begingroup$
You get $2$ what?
$endgroup$
– N. F. Taussig
Jan 23 at 14:22
$begingroup$
Typesetting hint: to display dollar signs, put a backslash in front:
$5 gives $5. Otherwise, everything between your dollar signs is interpreted as a mathematical expression.$endgroup$
– Théophile
Jan 23 at 14:23
$begingroup$
Typesetting hint: to display dollar signs, put a backslash in front:
$5 gives $5. Otherwise, everything between your dollar signs is interpreted as a mathematical expression.$endgroup$
– Théophile
Jan 23 at 14:23
1
1
$begingroup$
I think you mean $+$ instead of $*$ in your last sentence. Otherwise, you have exactly written out how to calculate the expected profit of the game. It's unclear what you mean by "the sum of the award of winning" in you second question though.
$endgroup$
– kccu
Jan 23 at 14:33
$begingroup$
I think you mean $+$ instead of $*$ in your last sentence. Otherwise, you have exactly written out how to calculate the expected profit of the game. It's unclear what you mean by "the sum of the award of winning" in you second question though.
$endgroup$
– kccu
Jan 23 at 14:33
2
2
$begingroup$
You should have addition signs rather than multiplication signs between each pair of parentheses since the three events are mutually exclusive.
$endgroup$
– N. F. Taussig
Jan 23 at 14:40
$begingroup$
You should have addition signs rather than multiplication signs between each pair of parentheses since the three events are mutually exclusive.
$endgroup$
– N. F. Taussig
Jan 23 at 14:40
1
1
$begingroup$
This is what happens when you don't take the trouble to ensure that your question is reasonably formatted: someone comes along and formats it for you, and while doing so they change a $+$ to $times$ and make it slightly worse (in that respect) than it was before. I recommend you go over the edited post and make sure everything there is exactly how you want it--notably those multiplication signs.
$endgroup$
– David K
Jan 23 at 14:40
$begingroup$
This is what happens when you don't take the trouble to ensure that your question is reasonably formatted: someone comes along and formats it for you, and while doing so they change a $+$ to $times$ and make it slightly worse (in that respect) than it was before. I recommend you go over the edited post and make sure everything there is exactly how you want it--notably those multiplication signs.
$endgroup$
– David K
Jan 23 at 14:40
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Your computation of the expected return is correct. Probably the sum to award question means to replace $2$ with some number to make the expected return $0$. Just replace $2$ by $x$ in your expected return expression, set it to zero, and solve the resulting equation.
answered Jan 23 at 14:41
Ross MillikanRoss Millikan
299k24200374
$endgroup$
add a comment |
$begingroup$
You are almost right. But recalling the expectation definition : $E(X) = sum p(X) X$ you should rather write
$$ frac{1}{4} times ($5+$2+$2- $5) + frac{1}{2} times ($5+$2- $5) + frac{1}{4}times ($0 - $5) $$
answered Jan 23 at 14:42
leonbloyleonbloy
41.5k647108
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your computation of the expected return is correct. Probably the sum to award question means to replace $2$ with some number to make the expected return $0$. Just replace $2$ by $x$ in your expected return expression, set it to zero, and solve the resulting equation.
answered Jan 23 at 14:41
Ross MillikanRoss Millikan
299k24200374
$endgroup$
add a comment |
$begingroup$
Your computation of the expected return is correct. Probably the sum to award question means to replace $2$ with some number to make the expected return $0$. Just replace $2$ by $x$ in your expected return expression, set it to zero, and solve the resulting equation.
answered Jan 23 at 14:41
Ross MillikanRoss Millikan
299k24200374
$endgroup$
add a comment |
$begingroup$
Your computation of the expected return is correct. Probably the sum to award question means to replace $2$ with some number to make the expected return $0$. Just replace $2$ by $x$ in your expected return expression, set it to zero, and solve the resulting equation.
answered Jan 23 at 14:41
Ross MillikanRoss Millikan
299k24200374
$endgroup$
Your computation of the expected return is correct. Probably the sum to award question means to replace $2$ with some number to make the expected return $0$. Just replace $2$ by $x$ in your expected return expression, set it to zero, and solve the resulting equation.
answered Jan 23 at 14:41
Ross MillikanRoss Millikan
299k24200374
answered Jan 23 at 14:41
Ross MillikanRoss Millikan
299k24200374
answered Jan 23 at 14:41
Ross MillikanRoss Millikan
299k24200374
answered Jan 23 at 14:41
Ross MillikanRoss Millikan
299k24200374
299k24200374
add a comment |
add a comment |
$begingroup$
You are almost right. But recalling the expectation definition : $E(X) = sum p(X) X$ you should rather write
$$ frac{1}{4} times ($5+$2+$2- $5) + frac{1}{2} times ($5+$2- $5) + frac{1}{4}times ($0 - $5) $$
answered Jan 23 at 14:42
leonbloyleonbloy
41.5k647108
$endgroup$
add a comment |
$begingroup$
You are almost right. But recalling the expectation definition : $E(X) = sum p(X) X$ you should rather write
$$ frac{1}{4} times ($5+$2+$2- $5) + frac{1}{2} times ($5+$2- $5) + frac{1}{4}times ($0 - $5) $$
answered Jan 23 at 14:42
leonbloyleonbloy
41.5k647108
$endgroup$
add a comment |
$begingroup$
You are almost right. But recalling the expectation definition : $E(X) = sum p(X) X$ you should rather write
$$ frac{1}{4} times ($5+$2+$2- $5) + frac{1}{2} times ($5+$2- $5) + frac{1}{4}times ($0 - $5) $$
answered Jan 23 at 14:42
leonbloyleonbloy
41.5k647108
$endgroup$
You are almost right. But recalling the expectation definition : $E(X) = sum p(X) X$ you should rather write
$$ frac{1}{4} times ($5+$2+$2- $5) + frac{1}{2} times ($5+$2- $5) + frac{1}{4}times ($0 - $5) $$
answered Jan 23 at 14:42
leonbloyleonbloy
41.5k647108
answered Jan 23 at 14:42
leonbloyleonbloy
41.5k647108
answered Jan 23 at 14:42
leonbloyleonbloy
41.5k647108
answered Jan 23 at 14:42
leonbloyleonbloy
41.5k647108
41.5k647108
add a comment |
add a comment |
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$begingroup$
You get $2$ what?
$endgroup$
– N. F. Taussig
Jan 23 at 14:22
$begingroup$
Typesetting hint: to display dollar signs, put a backslash in front:
$5gives $5. Otherwise, everything between your dollar signs is interpreted as a mathematical expression.$endgroup$
– Théophile
Jan 23 at 14:23
1
$begingroup$
I think you mean $+$ instead of $*$ in your last sentence. Otherwise, you have exactly written out how to calculate the expected profit of the game. It's unclear what you mean by "the sum of the award of winning" in you second question though.
$endgroup$
– kccu
Jan 23 at 14:33
2
$begingroup$
You should have addition signs rather than multiplication signs between each pair of parentheses since the three events are mutually exclusive.
$endgroup$
– N. F. Taussig
Jan 23 at 14:40
1
$begingroup$
This is what happens when you don't take the trouble to ensure that your question is reasonably formatted: someone comes along and formats it for you, and while doing so they change a $+$ to $times$ and make it slightly worse (in that respect) than it was before. I recommend you go over the edited post and make sure everything there is exactly how you want it--notably those multiplication signs.
$endgroup$
– David K
Jan 23 at 14:40