A computer is programmed to play the game of 24 by independently repla

Question

https://gmatclub.com/forum/download/file.php?id=74999 A computer is programmed to play the game of 24 by independently replacing the symbols ▲ and @ in a given numerical expression with one of the operations +, -, x, or ÷ and then computing the result. If the result is 24, the computer "wins". The table above shows the probabilities with which the computer will replace the symbols ▲ and @ with the operations +, -, x, or ÷. What is the probability that the computer will win when given the expression (4▲2) x (3@1)? A. 0.05 B. 0.24 C. 0.25 D. 0.30 E. 0.33 Capture.PNG

MartyMurray · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=74999

A computer is programmed to play the game of 24 by independently replacing the symbols ▲ and @ in a given numerical expression with one of the operations +, -, x, or ÷ and then computing the result. If the result is 24, the computer "wins". The table above shows the probabilities with which the computer will replace the symbols ▲ and @ with the operations +, -, x, or ÷. What is the probability that the computer will win when given the expression (4▲2) x (3@1)?

A. 0.05
B. 0.24
C. 0.25
D. 0.30
E. 0.33

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Attachment:

Capture.PNG

The values of (4▲2) produced via replacing ▲ with the 4 operations are 6, 2, 8, and 2.

The values of (3@1) produced via replacing @ with the four operations are 4, 2, 3, and 3.

A quick scan of the values reveals that, of the possible combinations produced when (4▲2) x (3@1) is replaced, only  6 × 4 and 8 × 3 will yield 24.

There's one way to produce 6 × 4, which is by replacing both symbols with +. The probability of replacing both with + is 0.40 × 0.60 = 0.24.

There are two ways to produce 8 × 3.

One is to replace @ with × and ▲ with ×. The probability of doing that is 0.20 × 0.25 = 0.05.

The other is to replace @ with × and ▲ with ÷. The probability of doing that is 0.20 × 0.05 = 0.01

So, the total probabliity of producing 24 is 0.24 + 0.05 + 0.01 = 0.30

A. 0.05
B. 0.24
C. 0.25
D. 0.30
E. 0.33

Correct answer:  D

egmat · Answer

This is such an amazing question with multiple layers of complexity. Watch the solution and determine which layer of complexity stumped you.

https://www.youtube.com/watch?v=L7sjSQILFCM

Complexity layer 1 – Understanding the scene wherein the symbols are mapped to one of the 4 operations. 
 Complexity layer 2 – Understanding that there is a probability associated with the translation of the symbols to each of the 4 operations. 
 Complexity layer 3 – Infering how the computer can win – i.e., how the operations can lead to a 24. One needs to arrive at multiple cases from this inference. 
 Complexity layer 4 – Infering which cases are possible. 
 Complexity layer 5 – Considering all cases in which a certain combination will work. 
 Complexity layer 6 – Reading appropriate values from the table and doing final calculations of the probability equation.

Here are the corrective actions based on which layer you stumbled on:
 
 Layers 1 or 2: Translation skill – you need to slow down while reading the dataset and you need to read the dataset with the intent to ‘own the dataset’. 
 Layers 3 or 4 – Inference skill – you need to hone your inference skills. 
 Layer 5 – Consider all cases – you need to ensure that you do not miss any cases. So consciously think about all the cases that can result in the output. 
 Layer 6 – Translation skill – you need to slow down and extract and process the data appropriately.

Happy learning!

Bunuel · Answer

Firstly, note that both (4▲2) and (3@1) will be positive integers no matter which arithmetic operations ▲ and @ represent. Next, 24 can be broken into the product of two positive integers in the following way: 1. 1 * 24 2. 2 * 12 3. 3 * 8 4. 4 * 6 5. 6 * 4 6. 8 * 3 7. 12 * 2 8. 24 * 1 Neither (4▲2) nor (3@1) can be 1 for any of the operations, so the 1st and 8th cases are out. (4▲2) also cannot be 3, 4, or 12. Hence, cases 3, 4, and 7 are also out. (3@1) cannot be 12. Hence, case 2 is also out. 1. 1 * 24 2. 2 * 12 3. 3 * 8 4. 4 * 6 5. 6 * 4 6. 8 * 3 7. 12 * 2 8. 24 * 1 (4▲2) is 6 when ▲ is addition. The probability of this is given to be 0.4. (3@1) is 4 when @ is addition. The probability of this is given to be 0.6. (4▲2) is 8 when ▲ is multiplication. The probability of this is given to be 0.2. (3@1) is 3 when @ is division OR multiplication. The probability of this is given to be 0.05 + 0.25. Therefore, the probability that (4▲2) x (3@1) is 24 is 0.4 * 0.6 + 0.2 * (0.05 + 0.25) = 0.3. Answer: D. GMAT-Club-Forum-o0maboda.png

KarishmaB · Answer

Because (4▲2) x (3@1) is a product of two factors, first of all, look for the product of two factors that give us 24.

1*24, 2*12, 3*8 and 4*6

Neither term can be made made more than 8 (= 4*2) hence ignore the first two products.

(4 * 2) x (3 * 1) = 8 * 3
(4 * 2) x (3 ÷ 1) = 8 * 3
(4 + 2) x (3 + 1) = 6 * 4

Probability of converting ▲ to x AND of converting  to  x or ÷ will be = 0.2 * 0.3 = 0.06
Probability of converting ▲ to + AND of converting  to  + will be = 0.4 * 0.6 = 0.24

Total probability = 0.06 + 0.24 = 0.3

Answer (D)

DanTheGMATMan · Answer

One of the best problems I've seen in a while. Just list out the possibilities:

https://www.youtube.com/watch?v=YD174rPFQvU

hemanthPA · Answer

Given: (4_2) * (3_1) = 24?

(4 x 2) * (3 x 1) = 8 x 3 = 24
(4 x 2) * ( 3 ÷ 1) = 8 x 3 = 24
(4 + 2) * (3 + 1) = 6 x 4 = 24

For 1 = 0.2 x 0.25 = 0.05
For 2 = 0.2 x 0.05 = 0.01
For 3 = 0.6 x 0.4 = 0.24

Total = 1 or 2 or 3 ways = 0.05 + 0.01 + 0.24 = 0.06 + 0.24 = 0.3

skatsaros · Answer

big thing with this is staying organized and structured.

GMATinsight · Answer

Quite a good one actually. To prepare for such questions, I share one method with all my students stating that even though you know the formula you must practice writing factors by making pairs of numbers whose product results in the number whose factor we are finding
e.g.

24 = 
1*24
2*12
3*8
4*6

Now, we need to think which case could be equated with  (4▲2) x (3@1)

(4▲2) as per our result could either be   4x2 = 8   or   4+2=6   or   4÷2 = 2   or 4-2 = 2 so, we have 4 cases here

Case 1:     (4▲2) = 4x2 = 8 the probability of this to happen is    0.2   
 in which case (3@1) must be 3 i.e. 3x1 or 3÷1 = 3   
 Probability of (3@1) to become   3x1 or 3÷1 = 0.25+0.05 =    0.30  
 So total Probability = 0.2*0.3 = 0.06

Case 2:     (4▲2) = 4+2 = 6 the probability of this to happen is    0.4  
 in which case (3@1) must be 4 i.e. 3+1 = 4   
 Probability of (3@1) to become 3+1 =    0.6  
 So total Probability = 0.4*0.6 = 0.24

Case 3:     (4▲2) = 4 ÷ 2 = 2 the probability of this to happen is    0.1  
 in which case (3@1) must be 12 BUT this is NOT POSSIBLE

Case 4:     (4▲2) = 4  -  2 = 2 the probability of this to happen is    0.3  
 in which case (3@1) must be 12 BUT this is NOT POSSIBLE

So Final Probrbaility = Case 1 + Case 2 = 0.06+0.24 = 0.3
ANswer: Option D

A computer is programmed to play the game of 24 by independently repla

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