Question: A telephone number contains 10 digit, including a 3-digit area code. Bob remembers the area code and the next 5 digits of the number. He also remembers that the remaining digits are not 0, 1, 2, 5, or 7. If Bob tries to find the number by guessing the remaining digits at random, the probability that he will be able to find the correct number in at most 2 attempts is closest to which of the following ?

1. 1/625
2. 2/625
3. 4/625
4. 25/625
5. 50/625

Correct Answer: E
Solution and Explanation:
Approach Solution 1:
The total number of possibilities for the phone numbers is 25,
Therefore the probability of him getting on the first try is 1/25
Here's where I differ from other posters: I would say that the probability of him getting it right on the second try would be:
(24/25)(1/24)

This is because the probability of him getting it wrong on the first try is 24/25, because there are 24 wrong answers and 1 right one. After that, however, he's already eliminated one possible wrong answer by trying and failing, so the total number of possibilities is now 24. That means he has a 1/24 chance of getting it right after trying one and failing.

This makes the total probability 1/25+1/25, which is exactly 50/625

Approach Solution 2:
Remaining numbers to fill last two digits (3,4,6,8,9): Total 5

Probability of choosing right numbers in two places = 1/5 * 1/5 = 1/25
Probability of not choosing right numbers in two places = 1-1/25 = 24/25

At most two attempts: 1) Wrong-1st Attempt, Right - 2nd, 2) Right - 1st Attempt
1) = 24/25 * 1/25 = 24/625
2) = 1/25

Add 1 and 2, 24/625 + 1/25 = 49/625 ~ 50/625

Approach Solution 3:
We see that each of the remaining 2 digits is 3, 4, 6, 8 or 9.

The probability he can guess the remaining 2 digits correctly in the first attempt is 1/5 x 1/5 = 1/25.

The probability he can guess the remaining 2 digits correctly in the second attempt (provided that he guessed them incorrectly in the first attempt) is (1 - 1/25) x 1/25 = 24/25 x 1/25 = 24/625.

Therefore, the probability he can guess the remaining 2 digits correctly in at most 2 attempts is 1/25 + 24/625 = 25/625 + 24/625 = 49/625 ≈ 50/625.

“A telephone number contains 10 digit, including a 3-digit”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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