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805+ Level|   Probability|         
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GMAT Club World Cup 2022 (DAY 3): Messi and Ronaldo are playing chess 13 Jul 2022, 08:00
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95% (hard)

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34% (02:11) correct 66% (02:32) wrong based on 156 sessions

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Messi and Ronaldo are playing chess till one of them wins a total of four games. If each of them is equally likely to win any game and no game ends in a draw, what is the probability that at least six games will be played?

(A) 3/8
(B) 7/16
(C) 1/2
(D) 9/16
(E) 5/8


 


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GMAT Club World Cup 2022 (DAY 3): Messi and Ronaldo are playing chess 13 Jul 2022, 09:02
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Solve it with the basic concept of Permutation and Probability.

Probability (P) that a game will be won by Messi- 1/2, same for Ronaldo- 1/2.

Now, one of them should win a total of four games to become a winner.

Probability 1= Winner will be decided in 4 games (case I) + Winner will be decided in 5 games (case II) + Winner will be decided in more than 5 games (i.e. at least 6 games- case III) ......... Equation 1

Case I- W W W W. Let's assume Messi wins all 4 starting game. Probability = 1/2 * 1/2 * 1/2 *1/2. Also Ronaldo has the same chances to win. So Answer for case I = 1/2 * 1/2 * 1/2 *1/2 * 2 = 1/8

Case II- L W W W W, W L W W W, W W L W W, W W W L W. Again let's assume Messi wins 4 & Ronaldo win 1game. Probablity = Probability = 1/2 * 1/2 * 1/2 *1/2 *1/2 * 4. Also Ronaldo will have the same chance. So Answer for case II = 1/2 * 1/2 * 1/2 *1/2 * 1/2 * 4 * 2 = 1/4

From Equaiton 1, Case III = 1- case I- caseII = 1- 1/8 - 1/4 = 5/8 (Option E)
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GMAT Club World Cup 2022 (DAY 3): Messi and Ronaldo are playing chess 13 Jul 2022, 08:32
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so atleast six games 3 should win messi and 3 ronaldo

total cases=2^6
favourable case=6(factorial)/3(fac)*3(fac)=20
probability=20/32=5/8 ansE
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GMAT Club World Cup 2022 (DAY 3): Messi and Ronaldo are playing chess 13 Jul 2022, 08:57
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In order to have both the players going into the sixth game, one of the players must win exactly three out of the first 5 games played which can be in any order- and may win or not win the 6th game ( note the keyword- "Atleast" )

Hence:
P= probability that one team wins only 3 matches in the first 5 / total number of possible combinations of wins
p= 5C3 * ([1][/2])^5 = 5/8
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GMAT Club World Cup 2022 (DAY 3): Messi and Ronaldo are playing chess 13 Jul 2022, 09:58
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Messi and Ronaldo are playing chess till one of them wins a total of four games. If each of them is equally likely to win any game and no game ends in a draw, what is the probability that at least six games will be played?

(A) 3/8
(B) 7/16
(C) 1/2
(D) 9/16
(E) 5/8


Games possible (to be payed) = 4,5,6,7 and so on.

Probability (atleast 6 games are payed) = 1 - probabilty ( 4 games OR 5 games)

P (4 games) = 2 * (0.5)^4 = 2/16 = 1/8 (Note. Multiplication by 2 to mean that either Messi or Ronaldo can be the winner)
P (4 games) = 4 * 2 * (0.5)^5 = 4/16 = 1/4 (Note. Multiplication by 4 since one game which is won by overall loser can be won either 1st , or 2nd, or 3rd or 4th game.It can't be the 5Th game; otherwise the match would have been finished in 4 games). Also, Multiplication by 2 to mean that either Messi or Ronaldo can be the winner )

So, Probability (atleast 6 games are payed) = 1 - (1/8 + 1/4)
= 1 - (3/8)
= 5/8

(E) is the CORRECT answer

Hope this helps..
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GMAT Club World Cup 2022 (DAY 3): Messi and Ronaldo are playing chess 14 Jul 2022, 06:54
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Bunuel
Messi and Ronaldo are playing chess till one of them wins a total of four games. If each of them is equally likely to win any game and no game ends in a draw, what is the probability that at least six games will be played?

(A) 3/8
(B) 7/16
(C) 1/2
(D) 9/16
(E) 5/8

 


This question was provided by GMAT Club
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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
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GMAT Club World Cup 2022 (DAY 3): Messi and Ronaldo are playing chess 18 Jun 2024, 05:10
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