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# A certain telephone number has seven digits. If the telephon

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Joined: 08 Oct 2009
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A certain telephone number has seven digits. If the telephon  [#permalink]

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Updated on: 09 Jul 2013, 10:51
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Difficulty:

65% (hard)

Question Stats:

64% (02:30) correct 36% (02:41) wrong based on 248 sessions

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A certain telephone number has seven digits. If the telephone number has the digit zero exactly 3 times, and the number 1 is not used at all, what is the probability that the phone number contains one or more prime digits.

A. 1/24
B. 1/16
C. 1/2
D. 15/16
E. 23/24

Originally posted by badgerboy on 14 Oct 2009, 05:00.
Last edited by Bunuel on 09 Jul 2013, 10:51, edited 1 time in total.
Renamed the topic and edited the question.
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Joined: 02 Sep 2009
Posts: 54401
Re: Telephone number with seven digits  [#permalink]

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15 Oct 2009, 17:08
3
2
cdowwe wrote:
Bunuel wrote:
bhushan252 wrote:
my approach...
after three zeros...we left with 4 places which can be filled with 2.3,4,5,6,7,8,9
this means 8 digits with 4 prime & 4 non prime nos to choose from.....

P = [(1P+3NP) OR (2P+2NP) OR (3P+1NP) OR (4P+0NP)]/[8C4]

[4C1*4C3+4C2*4C2+4C3*4C1+4C4*4C0] / [8C4]
[4*4+6*6+4*4+1]/70 = 69/70...BUT I AM WRONG SOMEWHERE

First step of getting rid of 3 digits of containing zeros is right: we are left with 4 digit number each of which can be one of 8 digit (2,3,4,5,6,7,8,9 as 0 is already used in first three and 1 is not used).

Next step:
total number of combinations possible 8^4,
combinations with NO PRIME 4^4,
P(no prime)=4^4/8^4=1/2^4
P(p>=1)=1-1/2^4=15/16

Answer D.

Can you explain the last step

Total # of combinations possible 8*8*8*8=8^4, as there are 8 digits available (2,3,4,5,6,7,8,9) for each of 4 digits.

Total # of combinations possible with no (meaning 0) primes 4*4*4*4=4^4 (as there are 4 non primes 4,6,8,9)

Probability that there will be 0 prime is 4^4/8^4=1/2^4

Probability at least one prime means that there will be 1,2 ,3 or 4 primes, so =1-probability of there will be 0 primes=1-1/2^4=15/16

Hope it's clear.
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Re: Telephone number with seven digits  [#permalink]

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14 Oct 2009, 05:19
my approach...
after three zeros...we left with 4 places which can be filled with 2.3,4,5,6,7,8,9
this means 8 digits with 4 prime & 4 non prime nos to choose from.....

P = [(1P+3NP) OR (2P+2NP) OR (3P+1NP) OR (4P+0NP)]/[8C4]

[4C1*4C3+4C2*4C2+4C3*4C1+4C4*4C0] / [8C4]
[4*4+6*6+4*4+1]/70 = 69/70...BUT I AM WRONG SOMEWHERE
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Re: Telephone number with seven digits  [#permalink]

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14 Oct 2009, 09:26
2
2
bhushan252 wrote:
my approach...
after three zeros...we left with 4 places which can be filled with 2.3,4,5,6,7,8,9
this means 8 digits with 4 prime & 4 non prime nos to choose from.....

P = [(1P+3NP) OR (2P+2NP) OR (3P+1NP) OR (4P+0NP)]/[8C4]

[4C1*4C3+4C2*4C2+4C3*4C1+4C4*4C0] / [8C4]
[4*4+6*6+4*4+1]/70 = 69/70...BUT I AM WRONG SOMEWHERE

First step of getting rid of 3 digits of containing zeros is right: we are left with 4 digit number each of which can be one of 8 digit (2,3,4,5,6,7,8,9 as 0 is already used in first three and 1 is not used).

Next step:
total number of combinations possible 8^,
combinations with NO PRIME 4^4,
P(no prime)=4^4/8^4=1/2^4
P(p>=1)=1-1/2^4=15/16

Answer D.
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Re: Telephone number with seven digits  [#permalink]

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15 Oct 2009, 16:07
Bunuel wrote:
bhushan252 wrote:
my approach...
after three zeros...we left with 4 places which can be filled with 2.3,4,5,6,7,8,9
this means 8 digits with 4 prime & 4 non prime nos to choose from.....

P = [(1P+3NP) OR (2P+2NP) OR (3P+1NP) OR (4P+0NP)]/[8C4]

[4C1*4C3+4C2*4C2+4C3*4C1+4C4*4C0] / [8C4]
[4*4+6*6+4*4+1]/70 = 69/70...BUT I AM WRONG SOMEWHERE

First step of getting rid of 3 digits of containing zeros is right: we are left with 4 digit number each of which can be one of 8 digit (2,3,4,5,6,7,8,9 as 0 is already used in first three and 1 is not used).

Next step:
total number of combinations possible 8^,
combinations with NO PRIME 4^4,
P(no prime)=4^4/8^4=1/2^4
P(p>=1)=1-1/2^4=15/16

Answer D.

Can you explain the last step
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Re: Telephone number with seven digits  [#permalink]

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05 May 2011, 07:11
1
Prime - 2,3,5,7

Non - Prime - 1,4,6,8

Format = 0 0 0 _ _ _ _

All Non-Primes = (4 * 4 * 4 * 4)/(8 * 8 * 8 * 8)

All Non-Primes = 1/16

At least 1 prime = 1 - 1/16 = 15/16

Answer - D
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Re: A certain telephone number has seven digits. If the telephon  [#permalink]

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10 Jan 2014, 08:02
badgerboy wrote:
A certain telephone number has seven digits. If the telephone number has the digit zero exactly 3 times, and the number 1 is not used at all, what is the probability that the phone number contains one or more prime digits.

A. 1/24
B. 1/16
C. 1/2
D. 15/16
E. 23/24

Yeah actually I failed to realize early in the problem that we didn't need to care about the 3 digits with the zero on it

But anyways, this is how I did it

What we are trying to do here is work in an inverse probabilistic approach hence, 1 - (Prob of having no primes)

Now, every other digit if the 4 that remain will have 8 possible options so total number of options is (8^4). This is total probabilities and denominator

Then for the numerator we can only take the numbers that aren't prime so 4,6,8,9. Everything else stays the same so we end up with (4^4)

Therefore, (4^4)/(8^4) = (1/2)^4 = 1/16

Now as we are looking for 1 - P(X) then answer will be 15/16

Hence D

Hope it helps
Cheers!
J
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Posts: 1661
Concentration: Finance
Re: Telephone number with seven digits  [#permalink]

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24 Feb 2014, 05:22
Bunuel wrote:
bhushan252 wrote:
my approach...
after three zeros...we left with 4 places which can be filled with 2.3,4,5,6,7,8,9
this means 8 digits with 4 prime & 4 non prime nos to choose from.....

P = [(1P+3NP) OR (2P+2NP) OR (3P+1NP) OR (4P+0NP)]/[8C4]

[4C1*4C3+4C2*4C2+4C3*4C1+4C4*4C0] / [8C4]
[4*4+6*6+4*4+1]/70 = 69/70...BUT I AM WRONG SOMEWHERE

First step of getting rid of 3 digits of containing zeros is right: we are left with 4 digit number each of which can be one of 8 digit (2,3,4,5,6,7,8,9 as 0 is already used in first three and 1 is not used).

Next step:
total number of combinations possible 8^,
combinations with NO PRIME 4^4,
P(no prime)=4^4/8^4=1/2^4
P(p>=1)=1-1/2^4=15/16

Answer D.

Hi Bunuel,

Just wanted to confirm something. Why is the order of the zeroes not important here? I mean why can we assume that zeroes will be the first 3 digits and not in any other order? I guess that the position of the zeroes does not change our probability scenario but just wanted to be 100% sure.

Thanks
Cheers
J
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Joined: 16 Oct 2010
Posts: 9125
Location: Pune, India
Re: A certain telephone number has seven digits. If the telephon  [#permalink]

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28 Apr 2016, 22:26
jlgdr wrote:
Bunuel wrote:
bhushan252 wrote:
my approach...
after three zeros...we left with 4 places which can be filled with 2.3,4,5,6,7,8,9
this means 8 digits with 4 prime & 4 non prime nos to choose from.....

P = [(1P+3NP) OR (2P+2NP) OR (3P+1NP) OR (4P+0NP)]/[8C4]

[4C1*4C3+4C2*4C2+4C3*4C1+4C4*4C0] / [8C4]
[4*4+6*6+4*4+1]/70 = 69/70...BUT I AM WRONG SOMEWHERE

First step of getting rid of 3 digits of containing zeros is right: we are left with 4 digit number each of which can be one of 8 digit (2,3,4,5,6,7,8,9 as 0 is already used in first three and 1 is not used).

Next step:
total number of combinations possible 8^,
combinations with NO PRIME 4^4,
P(no prime)=4^4/8^4=1/2^4
P(p>=1)=1-1/2^4=15/16

Answer D.

Hi Bunuel,

Just wanted to confirm something. Why is the order of the zeroes not important here? I mean why can we assume that zeroes will be the first 3 digits and not in any other order? I guess that the position of the zeroes does not change our probability scenario but just wanted to be 100% sure.

Thanks
Cheers
J

The question involves probability. The number of arrangements that you make in the restricted case (using no primes in the numerator) will be the same as the number of arrangements you make when you have no restrictions (all 8 digits possible in the denominator). Hence, you don't need to worry about the arrangements even though a telephone number depends on the order of the digits.

With three 0s taking away 3 places, you are left with 8 digits - 4 prime and 4 non prime
2, 3, 5, 7 and 4, 6, 8, 9
Probability of not using any prime for all 4 digits = (4/8)*(4/8)*(4/8)*(4/8) = 1/16
Probability of using at least one prime = 1 - 1/16 = 15/16
Answer (D)
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Re: A certain telephone number has seven digits. If the telephon  [#permalink]

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21 Jul 2017, 04:51
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Re: A certain telephone number has seven digits. If the telephon   [#permalink] 21 Jul 2017, 04:51
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# A certain telephone number has seven digits. If the telephon

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