Rachel and the Hustler
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 Mark Kudlowski
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Rachel and the Hustler
(I'd posed this as a GCSE question.)
Rachel is in Vegas, and notices a hustler's stall outside one of the casinos.
Hustler: "There are two normal pennies in here, an I'm offering 21 on you getting two heads when tossed."
Rachel: "I refuse those odds. They're biased in your favour."
Hustler: "There are three different results, two heads, two tails, or a head and a tail. So 21 is fair."
Rachel: "Right then. Can you offer me 21 on a head and a tail, then ? "
(Hustler thinks and diverts the question)
Hustler: "All right. There are ten normal pennies in here, and I'll give you the same 21 odds for getting 5 heads out of 10. I mean, half of 10 is 5, so I should only be offering you evens."
Rachel: "No. The chance of getting 5 heads out of 10 is even lower than getting 2 heads out of 2."
Show that Rachel was correct on both games.
Rachel is in Vegas, and notices a hustler's stall outside one of the casinos.
Hustler: "There are two normal pennies in here, an I'm offering 21 on you getting two heads when tossed."
Rachel: "I refuse those odds. They're biased in your favour."
Hustler: "There are three different results, two heads, two tails, or a head and a tail. So 21 is fair."
Rachel: "Right then. Can you offer me 21 on a head and a tail, then ? "
(Hustler thinks and diverts the question)
Hustler: "All right. There are ten normal pennies in here, and I'll give you the same 21 odds for getting 5 heads out of 10. I mean, half of 10 is 5, so I should only be offering you evens."
Rachel: "No. The chance of getting 5 heads out of 10 is even lower than getting 2 heads out of 2."
Show that Rachel was correct on both games.

 Acolyte
 Posts: 113
 Joined: Sat May 31, 2014 4:39 pm
 Location: Bournemouth
Re: Rachel and the Hustler
Based on the rules of probability every time you toss a coin it's an independent event. You have a 1 in 2 chance of hitting a head or tail each time. So to hit 2 heads it would be 1/2*1/2 equalling 1/4 which is odds of 41.
For the 5 heads from ten coins the best chance is 1/2*1/2*1/2*1/2*1/2 equalling 1/32 or 321 in terms of odds. However, that relies on hitting 5 heads in the first 5 goes. If that didn't happen the odds would go up in this order: 641; 1281; 2561; 5121; 10241.
I hope that's right
For the 5 heads from ten coins the best chance is 1/2*1/2*1/2*1/2*1/2 equalling 1/32 or 321 in terms of odds. However, that relies on hitting 5 heads in the first 5 goes. If that didn't happen the odds would go up in this order: 641; 1281; 2561; 5121; 10241.
I hope that's right

 Enthusiast
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 Joined: Wed Sep 18, 2013 2:32 pm
Re: Rachel and the Hustler
My understanding is that a 1 in 32 chance would be classed as 311 in terms of fair odds.
Do we mean exactly 5 heads or at least 5? For exactly 5 then the probability would be 0.5 to the power of 10, multiplied by the number of different ways this can be achieved eg the heads might be tosses 1 2 3 4 5 or tosses 2 3 6 7 9 etc etc. At school we used Pascal's triangle to arrive at this number. Can't be arsed checking it now but that's the principle anyway.
Do we mean exactly 5 heads or at least 5? For exactly 5 then the probability would be 0.5 to the power of 10, multiplied by the number of different ways this can be achieved eg the heads might be tosses 1 2 3 4 5 or tosses 2 3 6 7 9 etc etc. At school we used Pascal's triangle to arrive at this number. Can't be arsed checking it now but that's the principle anyway.

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 Posts: 271
 Joined: Wed Sep 18, 2013 2:32 pm
Re: Rachel and the Hustler
Change of heart. I now can be arsed.
The relevant number from the triangle is 252, so the probability of exactly 5 heads out of 10 is 24.61%.
I therefore assume this is what the original question was getting at, as the probability of at least 5 is just over 62%.
The relevant number from the triangle is 252, so the probability of exactly 5 heads out of 10 is 24.61%.
I therefore assume this is what the original question was getting at, as the probability of at least 5 is just over 62%.
 Mark Kudlowski
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 Posts: 433
 Joined: Fri Sep 12, 2008 3:15 pm
Re: Rachel and the Hustler
Fred Mumford wrote:Change of heart. I now can be arsed.
The relevant number from the triangle is 252, so the probability of exactly 5 heads out of 10 is 24.61%.
I therefore assume this is what the original question was getting at, as the probability of at least 5 is just over 62%.
Rachel was correct in both cases :
The prob. of 2 heads out of 2 is 1/4, so the true odds would be 31 against, not 21.
As a head and a tail can be made in 2 ways out of a possible 4, the true odds would be evens,
which is why the hustler changed his mind when Rachel suggested 21 for a head and a tail.
As for the case of 5 heads in 10, the Pascal's triangle entry is 252, so the prob. of exactly 5 heads in 10 is 252/1024 or 63/256, which is very slightly less than 1 in 4. The fair odds would be 193 to 63, or about 3.06 to 1.
This is an example of binomial distribution, for all statisticians among you.

 Acolyte
 Posts: 113
 Joined: Sat May 31, 2014 4:39 pm
 Location: Bournemouth
Re: Rachel and the Hustler
I was quite wrong with my answer then Great question though Mark
Re: Rachel and the Hustler
This has to go down as the most disappointing thread ever...
 Ian Volante
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Re: Rachel and the Hustler
I recommend the recent one about the penis programme then  it easily surpasses this one.sean d wrote:This has to go down as the most disappointing thread ever...
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