Rachel and the Hustler
Moderator: Michael Wallace
 Mark Kudlowski
 Enthusiast
 Posts: 430
 Joined: Fri Sep 12, 2008 3:15 pm
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
 Posts: 267
 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.

 Enthusiast
 Posts: 267
 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
 Enthusiast
 Posts: 430
 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
 Postmaster General
 Posts: 3311
 Joined: Wed Sep 03, 2008 8:15 pm
 Location: Edinburgh
 Contact:
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...
meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles
Who is online
Users browsing this forum: No registered users and 1 guest