Zen and the art of the bad run......

[Schizo]

I saw louie post this:"If you are routinely a 2:1 favorite against her you can EXPECT to lose 4 hands in a row against her every 56 hands"

Is that true? If so, how do you calculate that?

[elitegimp]

if you are a 2:1 favorite, you win 2/3 of the time. Therefore you lose 1/3 of the time -- the odds of losing 4 in a row is (1/3)^4 = 1/81.

So you expect to lose 4 in a row once every 81 times that you are a 2:1 favorite

[uuDevil]

[ QUOTE ]
if you are a 2:1 favorite, you win 2/3 of the time. Therefore you lose 1/3 of the time -- the odds of losing 4 in a row is (1/3)^4 = 1/81.

So you expect to lose 4 in a row once every 81 times that you are a 2:1 favorite

[/ QUOTE ]

But (I think) the probability of losing 4 or more in a row is (1/3)^4+(1/3)^5+(1/3)^6+(1/3)^7+....

If you approximate this by the first 3 terms, you get ~.0178 or ~1/56. (The actual value being closer to 1/54.)

[elitegimp]

[ QUOTE ]
[ QUOTE ]
if you are a 2:1 favorite, you win 2/3 of the time. Therefore you lose 1/3 of the time -- the odds of losing 4 in a row is (1/3)^4 = 1/81.

So you expect to lose 4 in a row once every 81 times that you are a 2:1 favorite

[/ QUOTE ]

But (I think) the probability of losing 4 or more in a row is (1/3)^4+(1/3)^5+(1/3)^6+(1/3)^7+....

If you approximate this by the first 3 terms, you get ~.0178 or ~1/56. (The actual value being closer to 1/54.)

[/ QUOTE ]

Interesting, but I'm not sure it's valid -- it's not possible to lose 5 in a row without losing 4 in a row so I don't think you can add those probabilities together (they're not independent).

That's clearly not a statistically valid response to your post, hopefully someone will figure out what I'm trying to say and say it for me /images/graemlins/smile.gif

Edit: and the infinite sum you mentioned above sums to (1/81)/(1-1/3) = 1/54 (first term / (1 - ratio between terms))

edit 2: it looks like you knew what I put in the first edit already, I didn't read through your post very well /images/graemlins/smile.gif

[BruceZ]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
if you are a 2:1 favorite, you win 2/3 of the time. Therefore you lose 1/3 of the time -- the odds of losing 4 in a row is (1/3)^4 = 1/81.

So you expect to lose 4 in a row once every 81 times that you are a 2:1 favorite

[/ QUOTE ]

But (I think) the probability of losing 4 or more in a row is (1/3)^4+(1/3)^5+(1/3)^6+(1/3)^7+....

If you approximate this by the first 3 terms, you get ~.0178 or ~1/56. (The actual value being closer to 1/54.)

[/ QUOTE ]

Interesting, but I'm not sure it's valid -- it's not possible to lose 5 in a row without losing 4 in a row so I don't think you can add those probabilities together (they're not independent).

[/ QUOTE ]

They aren't mutually exclusive, but it's OK because we are not computing the probability of losing 4 or more in a row. We are adding these probabilities to compute an expected value.

The probability of losing 4 in a row starting on a given hand is 1/81, but if he loses 5 in a row, he's counting that as 2 sequences of 4 in a row. If he loses 6 in a row, he counts that as 3 sequences of 4 in a row, etc. So in 81 hands, instead of losing 4 in a row an average of 1 time the way we normally count, instead we lose 4 in a row 1 time, then 1/3 of the time we lose another 1, then (1/3)^2 of the time we lose another 1, etc. So on average we lose 4 in a row

1 + 1/3 + (1/3)^2 + (1/3)^3 + ... = 1.5 times.

81 hands / 1.5 = 54 hands.

[elitegimp]

[ QUOTE ]


The probability of losing 4 in a row starting on a given hand is 1/81, but if he loses 5 in a row, he's counting that as 2 sequences of 4 in a row. If he loses 6 in a row, he counts that as 3 sequences of 4 in a row, etc. So in 81 hands, instead of losing 4 in a row an average of 1 time the way we normally count, instead we lose 4 in a row 1 time, then 1/3 of the time we lose another 1, then (1/3)^2 of the time we lose another 1, etc. So on average we lose 4 in a row

1 + 1/3 + (1/3)^2 + (1/3)^3 + ... = 1.5 times.

81 hands / 1.5 = 54 hands.

[/ QUOTE ]

Thanks, that clears a lot up!