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Probablity...Tough One

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Probablity...Tough One

by knight247 » Sun Sep 25, 2011 5:37 am
A lottery game works as follows: The player draws a numbered ball at random from an urn containing five balls numbered 1, 2, 3, 4, and 5. If the number on the ball is even, the player loses the game and receives no points; if the number on the ball is odd, the player receives the number of points indicated on the ball. Afterward, he or she replaces the ball in the urn and draws again. On each subsequent turn, the player loses the game if the total of the numbers becomes even, and gets another turn (after receiving the number of points indicated on the ball and then replacing the ball in the urn) each time the total remains odd.

(a) What is the probability that the player loses the game on the third turn?

(b) What is the probability that the player accumulates exactly 7 points and then loses on the next turn?
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by Brent@GMATPrepNow » Sun Sep 25, 2011 7:11 am
knight247 wrote:A lottery game works as follows: The player draws a numbered ball at random from an urn containing five balls numbered 1, 2, 3, 4, and 5. If the number on the ball is even, the player loses the game and receives no points; if the number on the ball is odd, the player receives the number of points indicated on the ball. Afterward, he or she replaces the ball in the urn and draws again. On each subsequent turn, the player loses the game if the total of the numbers becomes even, and gets another turn (after receiving the number of points indicated on the ball and then replacing the ball in the urn) each time the total remains odd.

(a) What is the probability that the player loses the game on the third turn?

(b) What is the probability that the player accumulates exactly 7 points and then loses on the next turn?
This game can never continue beyond the second draw.

First draw: To keep the game going, the first number drawn must be odd. Otherwise the game is over (by the rule that says "If the number on the ball is even, the player loses the game and receives no points")

Second draw: Once we draw the second number, the game will be over. The second number will be either odd or even. Consider each case:

case 1) we draw an even number, which means the game is over (by the rule that says "If the number on the ball is even, the player loses the game and receives no points")

case 2) we draw an odd number, which means the sum of our two drawn numbers is even (since odd + odd = even). At this point, the game is over (by the rule that says "the player loses the game if the total of the numbers becomes even")

Since the game can never move past the second draw, we can say that the both probabilities are equal to zero.

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by knight247 » Sun Sep 25, 2011 7:23 am
@Brent
Bro, I think you misunderstood the question. The person loses on getting an even only on the 1st turn. If he gets an even on the second turn its OK as long as the sum of his points is ODD.
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by Brent@GMATPrepNow » Sun Sep 25, 2011 7:33 am
knight247 wrote:@Brent
Bro, I think you misunderstood the question. The person loses on getting an even only on the 1st turn. If he gets an even on the second turn its OK as long as the sum of his points is ODD.
ahhhhhhhhhhhhhh,

Okay, then:

(a) What is the probability that the player loses the game on the third turn?

The only way for the player to lose on the 3rd turn is to get the following scenario:
Odd on first draw THEN even on second draw THEN odd on third draw (giving us an even sum and ending the game)

So, P(lose on 3rd turn) = P(odd on 1st draw AND even on 2nd draw AND odd on 3rd draw)
= P(odd on 1st draw) x P(even on 2nd draw) x P(odd on 3rd draw)
= 3/5 x 2/5 x 3/5
= 18/125

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by Brent@GMATPrepNow » Sun Sep 25, 2011 7:39 am
(b) What is the probability that the player accumulates exactly 7 points and then loses on the next turn?

Wayyyy too many calculations required to be a legitimate GMAT question.

In how many different ways can we accumulate 7 points and then lose on the next turn?
case a) 1-2-2-2-any odd
case b) 1-2-4-any odd
case c) 1-4-2-any odd
case d) 3-2-2-any odd
case e) 3-4-any odd
case f) 5-2-any odd

Yikes! Too many calculations here.
If someone wants to take it from here, fill your boots :-)

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by chetansharma » Mon Sep 26, 2011 8:20 am
knight247 wrote:@Brent
Bro, I think you misunderstood the question. The person loses on getting an even only on the 1st turn. If he gets an even on the second turn its OK as long as the sum of his points is ODD.
I could not get the highlighted point. Which statement has actually mentioned that the condition is valid only for the first turn? Please let me know if I have overlooked anything here . :oops:

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by Brent@GMATPrepNow » Mon Sep 26, 2011 8:26 am
chetansharma wrote:
knight247 wrote:@Brent
Bro, I think you misunderstood the question. The person loses on getting an even only on the 1st turn. If he gets an even on the second turn its OK as long as the sum of his points is ODD.
I could not get the highlighted point. Which statement has actually mentioned that the condition is valid only for the first turn? Please let me know if I have overlooked anything here . :oops:

Regards,
Chetan
Hi Chetan,

The original question is poorly worded. Nowhere does it specify that the condition is valid only for the first turn. This is a condition that knight247 later added to remove some of the ambiguity in the original question.

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by saketk » Mon Sep 26, 2011 8:44 am
Brent - to me the second part is certainly not a GMAT question. This looks more like the CAT (common admission test) exam held in India for IIMs ...

your thoughts ..
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by Brent@GMATPrepNow » Mon Sep 26, 2011 8:55 am
saketk wrote:Brent - to me the second part is certainly not a GMAT question. This looks more like the CAT (common admission test) exam held in India for IIMs ...
your thoughts ..
I think the math involved here is within the scope of the GMAT, but in my opinion, the question requires too many time-consuming calculations to be an official GMAT question. If students were given unlimited time to complete the test, then I think this question would be fine.

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by user123321 » Mon Sep 26, 2011 10:07 am
just giving a shot :)

1)What is the probability that the player loses the game on the third turn?
so he should win in first throw and second throw and then loose the next time.
(3c1/5c1)*(3c1/5c1)*(2c1/5c1) = 18/125

2)What is the probability that the player accumulates exactly 7 points and then loses on the next turn?
the possibilities will be...
1,1,1,1,1,1,1,(anyevennumber) = (1/5)^7*(2/5)
1,1,1,1,3,(anyevennumber) = 5*(1/5)^5*(2/5)
1,3,3,(anyevennumber) = 3*(1/5)^3*(2/5)
1,1,5,(anyevennumber) = 3*(1/5)^3*(2/5)
the answer will be sum of above individual probabilities.

edited:
user123321
Last edited by user123321 on Mon Sep 26, 2011 10:17 am, edited 1 time in total.
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by Brent@GMATPrepNow » Mon Sep 26, 2011 10:12 am
user123321 wrote: 2)What is the probability that the player accumulates exactly 7 points and then loses on the next turn?
the possibilities will be...
1,1,1,1,1,1,1 = (1/5)^7
1,1,1,1,3 = 5*(1/5)^5
1,3,3 = 3*(1/5)^3
1,1,5 = 3*(1/5)^3
the answer will be sum of above individual probabilities.
Sorry, but all of these break the rule that says "the player loses the game if the total of the numbers becomes even"

For example: 1, 3, 3
After the second draw, the sum of the selected number is 4 (which is even), so the game would end before we could select the second 3.

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by user123321 » Mon Sep 26, 2011 10:20 am
I am assuming that there is flaw in the question construction. because the second sentence says otherwise...
If the number on the ball is even, the player loses the game and receives no points;
I wish this should not come in gmat :)
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by Brent@GMATPrepNow » Mon Sep 26, 2011 10:24 am
user123321 wrote:I am assuming that there is flaw in the question construction. because the second sentence says otherwise...
If the number on the ball is even, the player loses the game and receives no points;
I wish this should not come in gmat :)
You are 100% correct; the wording of the question is very flawed.
If you read the entire thread, you will see that others have attempted to improve the question by adding additional text/rules.

Don't worry - in its original form, the question would never appear on the GMAT :-)

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