Probability: Danielle, Lilliana, Melody play fighting video game in tournament mode

Steven G

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Danielle, Lilliana, and Melody play a fighting video game in tournament mode. In this mode, two players play a match, and the winner of the match plays a new match against the player who was sitting out. This continues until a player wins two matches in a row. Danielle and Lilliana play the first match.
If they are all equally skilled at the game, then what is the probability that Melody will win the tournament?

Let's call the players D, L and M.

So D and L play and someone wins, say D.

So D and M plays. If D wins the tournament is over so M must win.

I claim that M must win the next round because if M loses (so L wins), then D and L plays. Whoever wins the D and L match won two matches and the tournament ends.

So, in the end M has to win two matches in a row which has a probability of 1/2 * 1/2 = 1/4.

My question is what is wrong with my logic/answer as 1/4 is not the correct answer?
 
Last edited:
One must also get to the match.

First, the Probability that there is a winner.

Round 1 0
Round 2 1/2
Round 3 1/4
Round 4 1/8
...
Round n 1/(2)^(n-1)

Second, the probability that the winner is Melody

Round 1 0 * 0
Round 2 0 * 1/2
Round 3 1/2 * 1/4
Round 4 0 * 1/8
Round 5 1/2 * 1/16
Round 6 0 * 1/32
Round 7 1/2 * 1/64
...

Looks like 2/5, to me.

Fascinating.
 
I misread the problem! I thought a player wins the tournament if they win 2 matches, not 2 matches in a row. I will try to solve this using the correct rules. For the record, the answer is 2/7.
 
One must also get to the match.

First, the Probability that there is a winner.

Round 1 0
Round 2 1/2
Round 3 1/4
Round 4 1/8
...
Round n 1/(2)^(n-1)

Second, the probability that the winner is Melody

Round 1 0 * 0
Round 2 0 * 1/2
Round 3 1/2 * 1/4
Round 4 0 * 1/8
Round 5 1/2 * 1/16
Round 6 0 * 1/32
Round 7 1/2 * 1/64
...

Looks like 2/5, to me. <== Bad!!

Fascinating.
Also an excellent lesson in paying attention. It only looked like 2/5 because someone tapped me on the shoulder and told me I was late for a meeting. Thus, the arithmetic never was checked. Eight minutes after the meeting, when my mind returned to more important things, I realized the errors. Now, it looks like 1/10. :)
 
Also an excellent lesson in paying attention. It only looked like 2/5 because someone tapped me on the shoulder and told me I was late for a meeting. Thus, the arithmetic never was checked. Eight minutes after the meeting, when my mind returned to more important things, I realized the errors. Now, it looks like 1/10. :)
Try again. I solved it a few minutes ago and did in fact get the answer of 2/7 which I was told was correct.
 
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