GMAT Club World Cup 2022 (DAY 3): Messi and Ronaldo are playing chess

1 - (No one wins all 4 OR No one wins 4 out or first 5 game)

A - MMMRR -> (½)^5 * 5!/(3!*2!) = 5/16
B - MMMM -> (½)^4 = 1/16

Hence, probability of required event = 1- ((5/16)+(1/16))
1-(6/16)
⅝
Option E is the correct choice

Note: ways to arrange is 5!/4!-1=4

1- [(1/2)^4*2+ (1/2)^5*2*4]=5/8

I did it without calculation just with observation !!

The max number of games that can be played is 7 (since after 4 wins of any player the game will end)
The min number of games that can be played is 4 (i.e, first four wins of a player)

Since there are equal chances of winning and losing a match, there is 50% probability that atleast 6 games will be played.

IMO Option C

Answer: C

The probability of first four games to be played will be 1.
For 5th game to be played, probability will be 1/2 since the game will be either played or not played. This will depend on the result whether anyone has win total four games by previous game.
Similarly, for any game to be played after 5th the probability will be 1/2 as the game will be either played or not played.
It won't matter if at least six games will be played or 90th game will be played.
Therefore, probability that at least six games will be played = 1/2

The answer is (D) 9/16

Getting answer as C

Just a wild Guess

IMO B. The clever guess)

Imo D

Probability that at least six games will be played = 1- (Probability of 1 game+ 2 games+ 3 games + 4games +5 games) ---- (1)

Probability that 1 or 2 or 3 game is played is 0 as they both need at least 4 games to win the game.
Probability that 4 games are played is that all are either won by Messi or Ronaldo = (1/ 2^4)+(1/2^4)
Probability that 5 games are played is that 4 won by messi and one loss by Messi and same for Ronaldo = ((5!/4!)/2^5 +(5!/4!)/2^5)

Putting above values in equation (1), we get Probability that at least six games will be played = 9/16

Given information:
1) Messi and Ronaldo are playing chess till one of them wins a total of 4 games.
2) Each likely to win
3) No draw
To determine: Probability(At least 6 games will be played)

Inferences:
1) Total possible options for a single game = 2 (Messi win or Ronaldo win, because no draw)
2) P(Either to win a game) = 1/2
3) P(At least 6 games) = 1 - Probability(Less than 6 games)

So, we need to actually determine P(Less than 6 games) {and subtract the result from 1} and minimum possible number of games for match to finish is 4 so we need to find the P(game finishing in 4 games) or P(game finishing in 5 games)
P(Less than 6 games) = P(4 games) + P(5 games)

P(4 games)

Case 1: Messi winning all
Total possible outcomes = 2*2*2*2 = 16
Favorable outcomes = 1
P = 1/16

Case 2: Ronaldo winning all
Total possible outcomes = 2*2*2*2 = 16
Favorable outcomes = 1
P = 1/16

Total = Messi winning all or Ronaldo winning all = 1/16 + 1/16 = 1/8

P(5 games)

Case 1: Messi winning 4 and Ronaldo winning 1 {RMMMM}
Total possible outcomes = 2*2*2*2*2 = 32
Total favorable outcomes = {5!/4!} - 1 = 4 [The reason for -1 is that there is one combination which will be MMMMR which means Messi wins the first 4 games but game will finish at that point so cannot have 5 games at that point and already covered 4 games scenario in previous part]
P = 4/32 = 1/8

Case 2: Ronaldo winning 4 and Messi winning 1 {MRRRR}
Total possible outcomes = 2*2*2*2*2 = 32
Total favorable outcomes = {5!/4!} - 1 = 4 [The reason for -1 is same as above]
P = 4/32 = 1/8

Total P = 2/8 = 2/8

Now: P(Less than 6) = P(4) + P(5) = 1/8 + 2/8 = 3/8

P(At least 6) = 1 - P(Less than 6) = 1 - (3/8) = 5/8

Answer - E

There are two possibilities with OR condition
1. Both the players have equal wins hence Messi wins 3 and Ronaldo wins 3
2. Either Messi or Ronaldo wins at least 2 games from the initial 5 games

Now basis of conditional probability we have
6C3 (1/2)^6 + 5C2 (1/2)^5
=> 20/2^6 + 20/2^6
=> 40/64 = 5/8

IMHO Option E

Favourable outcome=WWWLL (order does not matter)
So, 5!/3!2!=10
Since either Messi or Ronaldo may win, we need to multiply by 2 i.e. 10*2=20
Total outcomes=2^5=32
Probability=20/32=5/8
The answer is E.
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P(at least 6 games will be played)= 1-P(less than 6 games are played)

To get the result we need a minimum 4 games to be played
When four games are played, all the games should either be won by Messi or Ronaldo
P (result in 4 games)= 1/2*1/2*1/2*1/2*2= 1/8 (this is the result for either Messi or Ronaldo)

When 5 games are played, the following are the possibilities:
MRRRR, RMRRR, RRMRR, RRRMR
or
RMMMM, MRMMM, MMRMM, MMMRM
so 8 cases
P (result in 5 games)= 1/2*1/2*1/2*1/2*1/2*8= 1/4

P (result in 4 games)+ P(result in 5 games)= 1/8+1/4=3/8

P(atleast 6 games) 1-3/8=5/8

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