A lottery game works as follows: The player draws ...

[rasa.petrauskaite]

Source: Manhattan GMAT Guide 4 - Word Translations, 4th edition, P. 195, #6

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 all the numbers drawn 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.

(b) What is the probability that the player accumulates exactly 7 points and then loses on the next turn?


I have 2 questions about this problem:

1. The explanation of the problem on P. 199 makes sense to me. However, it would take me a long time to solve the problem and get the correct answer even if I used the method described there. Is there a faster method to solve this problem?

2. When I tried to solve the problem before looking at the answer, I used a different method. I wrote the following:

P(7 points and then lose) = x = ?

x = P(win on 1st turn) * (P(7 pts. on 2nd turn) + P(odd(not 7) on 2nd turn) * (P(7 pts. on 3rd turn) + P(odd(not 7) on 3rd turn) * (P(7 pts. on 4th turn) + ...))))

I also wrote out possible combinations of balls for each turn:

1+1=2
1+2=3
...

I calculated the probabilities written in the equation above.

x = (3/5)((2/15)+(4/15)((2/20)+(6/20)(1/25))) = 312/3,125

This answer is wrong. Where did I make a mistake?


Thank you for your time.

Rasa

[tim]

as to your first question, don't worry about the time it takes to answer In Action problems. they are not GMAT problems, so you do not need to worry about solving those in two minutes. several of those problems are designed to take longer than two minutes..

for your second question, can you explain where all the 15s and 20s come from in your denominators? that will help pinpoint your error, and you may even see the error yourself if you give some thought to where each of the numbers come from. my advice in general though would be to find a way to calculate all the different cases independently rather than using a string of nested parentheses. again, this may help you identify your error, and you also decrease the risk of creating a confusing mess..

[rasa.petrauskaite]

Sure. The 15s and 20s in my denominators come from the calculations of possible outcomes. During the first turn, there are 5 possible outcomes:
Ball 1, 2, 3, 4, or 5

On the second turn, there are 15 possible outcomes:
1 on first turn & 1 on second turn
1 on first turn & 2 on second turn
1 on first turn & 3 on second turn
etc.

On the third turn, there are 20 possible outcomes:
3 points sum for turns 1&2, and 1 on third turn
3 points sum for turns 1&2, and 2 on third turn
3 points sum for turns 1&2, and 3 on third turn
etc.

2 of the 20 possible outcomes yield 7 points.


I'm not sure if the last number in my equation should be 25 or 60. If I count 7, 9, and 11 totals from turns 1&2 once, I get 25 possible outcomes on turn 3. If I count them multiple times, I get 60. The reason I would want to count them multiple times is that there are multiple possibilities to attain those totals. For example, 5+4=9 and 7+2=9.

In ether case, the calculations yield a wrong answer.

x = (3/5)((2/15)+(4/15)((2/20)+(6/20)(1/60))) = 0.0968

The correct answer is 198/3125, which equals 0.06336.

I don't see my error, and I don't know why the above equation is not giving the correct answer.

Do you think I should count the totals multiple times in the case of the nested equation?

Thank you for your help.

Rasa

[ChrisB]

Rasa,

There are quite a few things amiss here.

1. Not accounting for both of the conditions in the desired outcome (rolling 7 AND losing next turn).
2. Not accounting for all outcomes (1,2,2,2,ODD is a desired outcome that takes 4 winning roles and 1 losing roll).
3. Determining all outcomes instead of saving time by sticking with independent probabilities

1. Desired outcomes: It is important to account for the game ending after one scores 7 points. Your model calculates the probability of scoring exactly 7, but not the game ending the next turn. For this reason, your answers are greater than the correct answer (e.g. 315/3125 vs. 198/3125).

To correct your answer you must account for rolling an odd number the turn after the turn in which 7 is scored. An extreme example of what you're doing is to rework the problem to test for the outcome where one rolls a 1 and loses on the next turn. You're calculating the answer as (1/5), which is the probability of rolling a 1 on turn one. To calculate the probability that you lose on turn 2 you must multiply by the probability of rolling an odd number because 1 + an odd = even. Thus the answer would be (1/5)x(3/5), not (1/5).

2. Not accounting for all outcomes - your model leaves out the case where one rolls 1,2,2,2,ODD. This is another outcome that fits the description.

3. Also, as Tim pointed out, each roll is independent so we'd prefer to see the associated probabilities with each roll have a denominator that's no greater than the total number of events (in this case 5). For example, in analyzing the two turn outcome you came up with 15 total outcomes, and found that 2 matched the constraints (2,5 and 5,2). Then you multiplied this number by (3/5) to account for being able to get to the second turn, resulting in (2/15)x(3/5). In this case, this product measures the probability of scoring 7 points and losing on turn 3, but does not measure the probability of scoring 7 points as you defined earlier.

To see why that's the case you must understand that you're double counting the fact that you've rolled an odd number in both outcomes. Imagine instead of 5 sides your die only had 2. Also imagine we wanted to calculate the probability of scoring 1 point and then the game ending. In that case there are 4 outcomes: 1,1 1,2 2,1 2,2. Only 1 matches the desired outcome so the answer is 1/4 and if you stopped there you'd be fine. But in your case, you multiplied 1/4 by 1/2, which is the probability of rolling a 1 on your first roll. This changes your answer to 1/8, which is incorrect. Thus, you get the wrong answer here as well as in your original solution, because you're multiplying a probability to account for an outcome that's already reflected in your answer.

As a takeaway, stick with the independent event model of the probability. It is more difficult to list out every outcome and as you pointed out, there's no way you would be able to solve a problem like this within two minutes using your method.

Regards,
Chris

[rasa.petrauskaite]

Thank you very much, Chris.

-Rasa

[jnelson0612]

Great discussion! Thanks all.