probability (1 Viewer)

[sam168]

cambridge y12 3u
chapter10h question 27

Eight players make the quarter-finals at wimbledon, the winer of each of the quarter-finals players a semi-final to see who enters the final.

a. assuming that all eight players are equally likely to win a catch, show that the probability that any tow particular players will play each other is 1/4

b. what is the probability that two people will play each other if the tournament strats with 16 players?

c. what is the probability that two particular players wil meet in a smilar konckout tornamentg if 2^n players enter?

Thank you in advance.

 

[braintic]

This is the 1982 4 unit Probability question.

If n=1, there are 2 players, so they MUST play: P=1

If n=2, there are 4 players.
If they are going to play, they must
EITHER:
Play in the first round: 1/3 chance, since each player has three possible opponents
OR:
Not play in the first round (2/3), both win their first round match [ (1/2)^2 = 1/4 ] AND play in the second round (1)
(2/3) * (1/4) * (1) = 1/6
P = (1/3) + (1/6) = 1/2

If n=3, there are 8 players.
If they are going to play, they must
EITHER:
Play in the first round: 1/7 chance, since each player has seven possible opponents
OR:
Not play in the first round (6/7), both win their first round match [ (1/2)^2 = 1/4 ] AND play in the remaining two rounds (1/2 - from n=2 case)
(6/7) * (1/4) * (1/2) = 3/28
P = (1/7) + (3/28) = 1/4

From this pattern, propose that the probability is 1/[2^(n-1)] and prove by induction:

If 2^(n+1) players, they must
EITHER:
Play in the first round: 1/[2^(n+1) - 1] chance, since each player has 2^(n+1) - 1 possible opponents
OR:
Not play in the first round [ (2^(n+1) - 2) / (2^(n+1) - 1) ], both win their first round match [ (1/2)^2 = 1/4 ] AND play in the remaining n rounds [ 1 / 2^(n-1) - from assumption for 2^n players]
[ (2^(n+1) - 2) / (2^(n+1) - 1) ] * (1/4) * [ 1 / 2^(n-1) ]

I'll leave it for you to add the two cases and finish off the algebra.

 

[sam168]

Genius!!! Thank you very much!

 

[sam168]

Geomietry

Triangle BAC angleABC=90 AB=AC. D and E on BC, BD=4 CE=3, F on AD and angleBFC=135, find the lenth of BF

 

[FrankXie]

Re: Geomietry

Triangle ABC angleABC=90 AB=AC. D and E on BC, BD=4 CE=3, F on AD and angleBFC=135, find the lenth of BF

I believe there were some typos: if angle ABC=90 degree, how can AB=AC? and since the point E seems irrelevant, i guess u missed something or should angle BFC be BFE?

 

[sam168]

Re: Geomietry

yes you are right. I made a mistake. Angle BAC=90

 

[FrankXie]

lol there should be one more typo