probability homework question, need some help

[jingramm]

Figured it out

 

[Gigantopithecus]

As you worded it, the problem doesn't make sense to me. Expected value that the match lasts two sets? Three sets? Variance of what? Match length? Even then, I don't know how you can calculate a variance without a mean, and how do you calculate a mean from a theoretically infinite set? (I.e. the match could last an infinite amount of sets.)

 

[Mr. Pedantic]

I'm bad at this kind of thing, but I got X = 2.5 and Var = 0.25

(I.e. the match could last an infinite amount of sets.)

Longest match length is 3 sets. Obviously.

 

[jingramm]

I'm bad at this kind of thing, but I got X = 2.5 and Var = 0.25


Longest match length is 3 sets. Obviously.

Yeah, the longest it would last is 3 sets. Can you show the math behind what you got?


Also, I think the 6 possible results are:
(Player 1, Player 1)
(Player 1, Player 2, Player 2)
(Player 1, Player 2, Player 1)
(Player 2, Player 2)
(Player 2, Player 1, Player 1)
(Player 2, Player 1, Player 2)

 

[jingramm]

I think I figured it out, does this look right?

With the 6 possible sets, I calculated it like this:

2*(2/6) + 3 (4/6) = 2.667


How about variance?

 

[Gigantopithecus]

I'm bad at this kind of thing, but I got X = 2.5 and Var = 0.25


Longest match length is 3 sets. Obviously.

Not if you know nothing about tennis, ha.

 

[TuxDave]

Not if you know nothing about tennis, ha.

So if I know NOTHING about tennis, I would think the longest match is NOT 3 matches?

 

[preslove]

Does it say that the players are women? Dudes play for 5 sets.

edit: oops . I saw Gigantopithecus's error and forgot about the "wins 2 sets" part.

 

[jingramm]

Does it say that the players are women? Dudes play for 5 sets.

huh?


It says the tennis match lasts until whoever wins two sets. The max that this would last is three. You can not go beyond three sets without a person winning two.

 

[jman19]

Not if you know nothing about tennis, ha.

Two players, and the match ends once one player has won two sets... you don't need to know anything about tennis beyond what the OP said.

 

[Gigantopithecus]

Two players, and the match ends once one player has won two sets... you don't need to know anything about tennis beyond what the OP said.

Yes, I am apparently a retard and read 'wins two sets' as 'wins two consecutive sets.'

 

[jman19]

Yes, I am apparently a retard and read 'wins two sets' as 'wins two consecutive sets.'

Fair enough :thumbsup:

 

[Mr. Pedantic]

Yeah, the longest it would last is 3 sets. Can you show the math behind what you got?

Each combination of a 2-set match has a 1/4 chance of occurring. Each combination of a 3-set match has a 1/8 chance of occurring. So while there are twice as many 3-set combinations as 2-set combinations, the 2-set combinations happen twice as often. This means half the time the match will have 2 sets, and half the time the match will have 3 sets. Or:

AA 1/4
ABA 1/8
ABB 1/8
BAB 1/8
BAA 1/8
BB 1/4

As for variance, I just got my calculator to figure it out.

Does it say that the players are women? Dudes play for 5 sets.

Most ATP events only score best of 3. Only the Slams and Davis Cup AFAIK score to best of 5.

 

[Narmer]

Each combination of a 2-set match has a 1/4 chance of occurring. Each combination of a 3-set match has a 1/8 chance of occurring. So while there are twice as many 3-set combinations as 2-set combinations, the 2-set combinations happen twice as often. This means half the time the match will have 2 sets, and half the time the match will have 3 sets. Or:

AA 1/4
ABA 1/8
ABB 1/8
BAB 1/8
BAA 1/8
BB 1/4

As for variance, I just got my calculator to figure it out.


Most ATP events only score best of 3. Only the Slams and Davis Cup AFAIK score to best of 5.

How do you have 6 combinations and say they are divisible by 8? If the game is over once you win 2 games then shouldn't it be divisible by 6, as in 1/3 and 2/3 as the OP did?

 

[Mr. Pedantic]

Work it out. Each of the 2-set combinations occurs at twice the probability of each of the 3-set combinations. And there are 8 combinations:

AAA
AAB
ABA
ABB
BAB
BAA
BBA
BBB

Each of these occur with a probability of 12.5%. However, because of the conditions of the scenario, AAA and AAB are lumped into one result - AA. Same with BBA and BBB. AA and BB then each have 12.5 x 2 = 25% probability in every given match, whereas the 3-set matches each have a 12.5% probability.

If you want to think of it another way: imagine it's the end of the first set. Someone (it doesn't matter who) has won it - let's call him A; and also let's call the loser B. There are two possibilities that could now occur. A could win the set, or B could win the set. If A wins (50% probability of this occurring) then the game is 2 sets long. If B wins (again, 50%) then the game will be 3 sets long.

If you really want to test it you can flip coins all day to work it out.

 

[Furyline]

Once the first set is over, there's a 50% chance that whoever won the first set will win the second set. So half the time it'll be 2 sets, half the time it'll be 3 sets.

Showing some math from earlier in the thread:
P(AA) = (1/2)*(1/2) = 1/4
P(ABA) = (1/2)*(1/2)*(1/2) = 1/8
...

 

[TecHNooB]

Two players play a tennis match, which ends when one of the players has won two sets. Suppose that each set is equally likely to be won by either player, and that the results from different sets are independent.

What is the expected value and what is the variance of the number of sets player?


I honestly tried solving on my own and searched the web but I'm still not clear on those two things.

AA (.25)
ABA (.125)
ABB (.125)
BB (.25)
BAB (.125)
BAA (.125)

E = 2*.25 + 2*.25 + 3*.125 + 3*.125 + 3*.125 + 3*.125 = 2.5
Var = E(X^2) - E(X)^2 = 4*.25 + 4*.25 + 9*.125 + 9*.125 + 9*.125 + 9*.125 - 2.5^2

 

[Narmer]

Work it out. Each of the 2-set combinations occurs at twice the probability of each of the 3-set combinations. And there are 8 combinations:

AAA
AAB
ABA
ABB
BAB
BAA
BBA
BBB

Each of these occur with a probability of 12.5%. However, because of the conditions of the scenario, AAA and AAB are lumped into one result - AA. Same with BBA and BBB. AA and BB then each have 12.5 x 2 = 25% probability in every given match, whereas the 3-set matches each have a 12.5% probability.

If you want to think of it another way: imagine it's the end of the first set. Someone (it doesn't matter who) has won it - let's call him A; and also let's call the loser B. There are two possibilities that could now occur. A could win the set, or B could win the set. If A wins (50% probability of this occurring) then the game is 2 sets long. If B wins (again, 50%) then the game will be 3 sets long.

If you really want to test it you can flip coins all day to work it out.

Thanks.

 

[preslove]

huh?


It says the tennis match lasts until whoever wins two sets. The max that this would last is three. You can not go beyond three sets without a person winning two.

oops . I saw Gigantopithecus's error and forgot about the "wins 2 sets" part.