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A telephone number contains 10 digit, including a 3-digit

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Senior Manager
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Joined: 12 Aug 2015
Posts: 280
Concentration: General Management, Operations
GMAT 1: 640 Q40 V37
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WE: Management Consulting (Consulting)
A telephone number contains 10 digit, including a 3-digit  [#permalink]

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New post 02 Nov 2015, 11:15
1
so Bob is given 2 opportunities to decipher the remaining 2 digits of the code, for this task he has 5 digits to choose from.

IMPORTANT notes to bear in mind before calculation:
- picking a number for place 9 is independent from picking the number for place 10 - i.e. digits can repeat
- if Bob wastes his 1st opportunity he will have better chances to pick the right combination during his second attempt - because by that time he has already proved that one combination is invalid.

So we have all the info needed to do simple calculation:

(1) \(\frac{1}{5}\)*\(\frac{1}{5}\) = \(\frac{1}{25}\) - this is the independent event that he would win on first attempt

OR

(2.1) \(\frac{4}{5}\)*\(\frac{4}{5} = [m]\frac{16}{25}\) - he fails to choose correctly on first attempt because he picks either of the 4 wrong digits for the both places

(2.2) \(\frac{1}{4}\)*\(\frac{1}{4}\)=\(\frac{1}{16}\) - he finally chooses the correct digit out of the remaining 4 for each place independently!

AND

(2.3) Multiply the above two iterations to complete the chances for the second scenario: \(\frac{16}{25}\)*\(\frac{1}{16}\)=\(\frac{1}{25}\)

(3) Sum up \(\frac{1}{25}\)+\(\frac{1}{25}\)=\(\frac{2}{25}\)=\(\frac{50}{625}\)
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Re: A telephone number contains 10 digit, including a 3-digit  [#permalink]

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New post 02 Jul 2019, 10:38
srp wrote:
I am getting E (as closest)

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


In case 1, when you attempted for the 1st time and got it wrong, 1 possibility out of 25 is exhausted. You are left with only 24 possibilities. So, the second attempts outcome should be measured on the remaining 24 possibilities only.

Then you will get the exact answer.
=(24/25)*(1/24)+(1/25)
=1/25 + 1/25
=2/25 which is same as 50/625.

Hope this helped.

Please correct me if I am wrong. Thank you.

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Re: A telephone number contains 10 digit, including a 3-digit   [#permalink] 02 Jul 2019, 10:38

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