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Intern  Joined: 30 Aug 2010
Posts: 13
A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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2
5 00:00

Difficulty:   55% (hard)

Question Stats: 67% (02:17) correct 33% (01:55) wrong based on 274 sessions

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A four digit safe code does not contain the digits 1 and 4 at all. What is the probability that it has at least one even digit?

a) ¼
b) ½
c) ¾
d) 15/16
e) 1/16
Intern  Joined: 30 Aug 2010
Posts: 13
Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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1
2
As already said in a previous post (out-of-a-box-that-contains-4-black-and-6-white-mice-101375.html), I'm experiencing some problems with probability and combinations. So to understand better these arguments, I need to solve the problem in several ways. Unfortunately approaches 2 and 4 are missing. Could you help me?

1) probability approach:
Probability of at least one even digit = Probability of one even digit + Probability of two even digits + Probability of three even digits + Probability of four even digits
I have four even digits: 0,2,6,8 and four odd digits: 3,5,7,9
So the probability of even digit is $$1/2$$, as well of odd digit.

P1 = $$1/2$$
P2 = $$1/2^2=1/4$$
P3 = $$1/2^3=1/8$$
P4 = $$1/2^4=1/16$$
P = P1+P2+P3+P4 = $$15/16$$

2) combinatorial approach:
This approach is missing.

3) reversal probability approach:
Probability of at least one even digit = 1 - Probability of four odd digits
P = $$1 - 1/2^4 = 15/16$$

3) reversal combinatorial approach:
This approach is missing.
Retired Moderator Joined: 02 Sep 2010
Posts: 759
Location: London
Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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2
Combinatorial Solution

No of ways to form code with atleast 1 even digit = Total Ways - No of ways to form code with only odd digits

The digits available are {5,6,7,8,9,0}

Total ways are 6x6x6x6 = 6^4

Ways using only odd digits = 3x3x3x3 = 3^4

Therefore Probability = $$\frac{6^4 - 3^4}{6^4} = 1 - \frac{3^4}{6^4} = 1 - (1/2)^4 = 15/16$$
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Intern  Joined: 30 Aug 2010
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Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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2
1
shrouded1 wrote:
Combinatorial Solution

No of ways to form code with atleast 1 even digit = Total Ways - No of ways to form code with only odd digits

The digits available are {5,6,7,8,9,0}

Total ways are 6x6x6x6 = 6^4

Ways using only odd digits = 3x3x3x3 = 3^4

Therefore Probability = $$\frac{6^4 - 3^4}{6^4} = 1 - \frac{3^4}{6^4} = 1 - (1/2)^4 = 15/16$$

Your approach is correct (even if I would call it reversal combinatorial approach) but with this number:
Digits available are {0,2,3,5,6,7,8,9,0}
Total ways are = 8x8x8x8 = $$8^4 = 2^{12}$$
Ways using only odd digits = 4x4x4x4 = $$4^4 = 2^8$$
Therefore probability = $$\frac{2^{12} - 2^8}{2^{12}} = 1 - \frac{2^8}{2^{12}} = 1 - (1/2)^4 = 15/16$$

Thanks a lot!
Retired Moderator Joined: 02 Sep 2010
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Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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oh now i see it, i thought you said none of the digits from 1 to 4, but as you pointed out the solution remains the same.

The "direct combinatorial" approach as you call it will be painful in this case. You will need to calculate each of 1 even, 2 even, 3 even, all even. The terms are themselves probably easy enough to calculate, but simplifying will be a bit tedious to get the same answer
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Location: London
Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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1
rraggio wrote:
As already said in a previous post (out-of-a-box-that-contains-4-black-and-6-white-mice-101375.html), I'm experiencing some problems with probability and combinations. So to understand better these arguments, I need to solve the problem in several ways. Unfortunately approaches 2 and 4 are missing. Could you help me?

1) probability approach:
Probability of at least one even digit = Probability of one even digit + Probability of two even digits + Probability of three even digits + Probability of four even digits
I have four even digits: 0,2,6,8 and four odd digits: 3,5,7,9
So the probability of even digit is $$1/2$$, as well of odd digit.

P1 = $$1/2$$
P2 = $$1/2^2=1/4$$
P3 = $$1/2^3=1/8$$
P4 = $$1/2^4=1/16$$
P = P1+P2+P3+P4 = $$15/16$$

I know you got to the answer, but your approach isnt quite correct.

P1 = Probability of exactly 1 even digit = (1/2)^4 * C(4,1) = 4/16
P2 = Probability of exactly 2 even digits = (1/2)^4 * C(4,2) = 6/16
P3 = Probability of exactly 3 even digits = (1/2)^4 * C(4,3) = 4/16
P4 = Probability of 4 even digits = 1/16
P = P1+P2+P3+P4 = 15/16

You have calculated P1 as prob of at least 1 even digit, P2 as atleast 2 even digits etc. This is incorrect. Its only coincidence that the answer is correct.
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GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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1
probability of at least one even = 1-(probability of none even)

probability of none even = required cases/total cases

total cases = 8*8*8*8 as 0 2 3 5 6 7 8 9 all 8 can be taken for each digit

required cases = 4*4*4*4 as only 3 5 7 9 can be taken

probability of none even = $$\frac{4*4*4*4}{(8*8*8*8)}= \frac{1}{16}$$

probability of at least one even =$$1-\frac{1}{16} = \frac{15}{16}$$
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Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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1
I solved the question thinking 1,2,3 and 4 are not allowed in safecode and got same answer . My ratios were
3*3*3*3/(6*6*6*6). So by coincidence I got same answer _________________
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Intern  Joined: 10 Oct 2010
Posts: 21
Location: Texas
Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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rraggio wrote:
A four digit safe code

Permutation
(_)(_)(_)(_)

Note: A safe code can have 0 as the thousands digit; a 4-digit number cannot.

rraggio wrote:
does not contain the digits 1 and 4 at all.

Bag of 8 choices w/ replacement.

rraggio wrote:
What is the probability that it has at least one even digit?

"At least" means Probability Table. Create and work backwards.

****************************
# of evens: Events
0:
1:
2:
3:
4:
----------------------------
Total =
****************************

Total = Fill in Permutation above = 8P4 = (8)(8)(8)(8)

4 evens:
4 evens pick 1 AND
4 evens pick 1 AND
4 evens pick 1 AND
4 evens pick 1 TIMES
the number of ways to arrange EEEE
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/4! = 4*4*4*4 * 1

3 evens:
4 evens pick 1 AND
4 evens pick 1 AND
4 evens pick 1 AND
4 odds pick 1 TIMES
the number of ways to arrange EEEO
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/3!1!= 4*4*4*4 * 4

2 evens:
4 evens pick 1 AND
4 evens pick 1 AND
4 odds pick 1 AND
4 odds pick 1 TIMES
the number of ways to arrange EEOO
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/2!2! = 4*4*4*4 * 6

1 even:
4 evens pick 1 AND
4 odds pick 1 AND
4 odds pick 1 AND
4 odds pick 1 TIMES
the number of ways to arrange EOOO
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/1!3! = 4*4*4*4 * 4

0 evens:
4 odds pick 1 AND
4 odds pick 1 AND
4 odds pick 1 AND
4 odds pick 1 TIMES
the number of ways to arrange OOOO
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/4! = 4*4*4*4 * 1

****************************

# of evens: Events
0: 4*4*4*4*1
1: 4*4*4*4*4
2: 4*4*4*4*6
3: 4*4*4*4*4
4: 4*4*4*4*1
----------------------------
Total = 8*8*8*8 = 4*4*4*4*2*2*2*2 =4*4*4*4*16

****************************

Use the table info to answer any questions.
P(evens = 0) = 1/16
P(evens > 0) = (4+6+4+1)/16
etc.

rraggio wrote:
a) ¼
b) ½
c) ¾
d) 15/16
e) 1/16

ANS: D
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Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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rraggio wrote:
A four digit safe code does not contain the digits 1 and 4 at all. What is the probability that it has at least one even digit?

a) ¼
b) ½
c) ¾
d) 15/16
e) 1/16

Total available digits to make 4 digit code= 8 (excludes 1 and 4)

Probability of having at least one even digit= 1- probability of having all odd digits

Total ways to make 4 digit code= 8*8*8*8
Total ways of having only odd digits= 4*4*4*4
probability of having all odd digits= 4*4*4*4/8*8*8*8=1/16

Probability of having at least one even digit= 1- 1/16= 15/16

D is the answer
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Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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The digits to be considered are 0,5,6,7,8,9 so the total is 6c4 =15 and having at least one even number is 4c1+4c2+4c3=14 Therefore the probability is 14/15. I know my approach is wrong . Can someone please tell me as to where am i going wrong ?
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Joined: 02 Sep 2009
Posts: 55267
Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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longhaul123 wrote:
The digits to be considered are 0,5,6,7,8,9 so the total is 6c4 =15 and having at least one even number is 4c1+4c2+4c3=14 Therefore the probability is 14/15. I know my approach is wrong . Can someone please tell me as to where am i going wrong ?

The question says that "...code does not contain the digits 1 and 4 at all", not "from 1 through 4". Meaning that we can use 2 and 3,
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Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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rraggio wrote:
A four digit safe code does not contain the digits 1 and 4 at all. What is the probability that it has at least one even digit?

a) ¼
b) ½
c) ¾
d) 15/16
e) 1/16

We can use the following equation:

P(at least one even digit) = 1 - P(no even digits)

Since 1 and 4 cannot be used, we have 8 available digits (0, 2, 3, 5, 6, 7, 8, 9), and we see that 4 of those 8 digits are odd (3,5,7,9). Thus, P(no even digits) = 4/8 x 4/8 x 4/8 x 4/8 = (1/2)^4 = 1/16.

Thus, P(at least one even digit) = 1 - 1/16 = 15/16.

Answer: D
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Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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How can all the 4 digits be filled with 8 combination. If it has to be a 4 digit number and 1,4 are not allowed then first digit of 4 digit number can only be filled by 7 numbers ( excepting 1,4, 0) , so shouldn't it be :

7*8*8*8

What am i missing here ?

cabelk wrote:
rraggio wrote:
A four digit safe code

Permutation
(_)(_)(_)(_)

Note: A safe code can have 0 as the thousands digit; a 4-digit number cannot.

rraggio wrote:
does not contain the digits 1 and 4 at all.

Bag of 8 choices w/ replacement.

rraggio wrote:
What is the probability that it has at least one even digit?

"At least" means Probability Table. Create and work backwards.

****************************
# of evens: Events
0:
1:
2:
3:
4:
----------------------------
Total =
****************************

Total = Fill in Permutation above = 8P4 = (8)(8)(8)(8)

4 evens:
4 evens pick 1 AND
4 evens pick 1 AND
4 evens pick 1 AND
4 evens pick 1 TIMES
the number of ways to arrange EEEE
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/4! = 4*4*4*4 * 1

3 evens:
4 evens pick 1 AND
4 evens pick 1 AND
4 evens pick 1 AND
4 odds pick 1 TIMES
the number of ways to arrange EEEO
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/3!1!= 4*4*4*4 * 4

2 evens:
4 evens pick 1 AND
4 evens pick 1 AND
4 odds pick 1 AND
4 odds pick 1 TIMES
the number of ways to arrange EEOO
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/2!2! = 4*4*4*4 * 6

1 even:
4 evens pick 1 AND
4 odds pick 1 AND
4 odds pick 1 AND
4 odds pick 1 TIMES
the number of ways to arrange EOOO
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/1!3! = 4*4*4*4 * 4

0 evens:
4 odds pick 1 AND
4 odds pick 1 AND
4 odds pick 1 AND
4 odds pick 1 TIMES
the number of ways to arrange OOOO
= 4C1 * 4C1 * 4C1 * 4C1 * 4!/4! = 4*4*4*4 * 1

****************************

# of evens: Events
0: 4*4*4*4*1
1: 4*4*4*4*4
2: 4*4*4*4*6
3: 4*4*4*4*4
4: 4*4*4*4*1
----------------------------
Total = 8*8*8*8 = 4*4*4*4*2*2*2*2 =4*4*4*4*16

****************************

Use the table info to answer any questions.
P(evens = 0) = 1/16
P(evens > 0) = (4+6+4+1)/16
etc.

rraggio wrote:
a) ¼
b) ½
c) ¾
d) 15/16
e) 1/16

ANS: D
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Posts: 7
Re: A Four digit safe code does not contain the digits 1 and 4..  [#permalink]

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Since repetition of digits is allowed, total number of ways 8 digits can be used = 2^8
Odd digits - 3,5,7,9
Even digits - 0,2,6,8
Ways in which all odd digits are used - 2^4
Probability of all odd digits being used = (2^4)/(2^8)=1/16
Probability of at least one even digit = 1-1/16=15/16 Re: A Four digit safe code does not contain the digits 1 and 4..   [#permalink] 01 Feb 2018, 00:16
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# A Four digit safe code does not contain the digits 1 and 4..

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