John wrote a phone number on a note that was later lost

The phone numbers is of a type: {X}{X}{X}{X}{1}{1}{1}.

{X}'s can take following values: 4 primes {2, 3, 5, 7} and 4 non-primes {4, 6, 8, 9}. Total 8 choices for each {X}. Probability that {X} will be prime is therefore \(\frac{4}{8}=\frac{1}{2}\) and probability of {X} will not be a prime is again \(\frac{1}{2}\).

We want at least 2 {X}'s out of 4 to be primes, which means 2, 3 or 4 primes.

Let's count the opposite probability and subtract it from 1.

Opposite probability of at least 2 primes is 0 or 1 prime:

So {P}{NP}{NP}{NP} and {NP}{NP}{NP}{NP}.

Scenario 1 prime - {P}{NP}{NP}{NP}: \(\frac{4!}{3!}*\frac{1}{2}*(\frac{1}{2})^3=\frac{4}{16}\). We are multiplying by \(\frac{4!}{3!}\) as scenario {P}{NP}{NP}{NP} can occur in several different ways: {P}{NP}{NP}{NP}, {NP}{P}{NP}{NP}, {NP}{NP}{P}{NP}, {NP}{NP}{NP}{P} - 4 ways (basically the # of permutations of 4 objects out ow which 3 are the same).

Scenario 0 prime - {NP}{NP}{NP}{NP}: \((\frac{1}{2})^4=\frac{1}{16}\).

Hence opposite probability = \(\frac{4}{16}+\frac{1}{16}=\frac{5}{16}\).

So probability of at least 2 primes is: 1-(Opposite probability) = \(1-\frac{5}{16}=\frac{11}{16}\)

Answer: B.
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Yes, it does seem that the language of the question is not clear. When I read the question, I also assumed 4 prime digits and 5 non prime (including 1). After all, the question only says that 1 appears in the last 3 places (and hence, we can ignore the last 3 places). It doesn't say that 1 does not appear anywhere else. But it seems that it might also imply that 1 appears only in the last 3 places (looking at the options, that was their intention). Anyway, I am sure that if this question actually appears and they mean to say that 1 cannot be at any other place, they will definitely mention it.
Let's calculate the probability of different number of prime numbers:

0 primes
Probability = (5/9)*(5/9)*(5/9)*(5/9)

1 prime
Probability = (4/9)*(5/9)*(5/9)*(5/9)*4 (there are 4 positions for the prime)

2 primes
Probability = (4/9)*(4/9)*(5/9)*(5/9)*4C2 (select 2 of the 4 positions for the primes)

3 primes
Probability = (4/9)*(4/9)*(4/9)*(5/9)*4 (4 positions for the non prime)

4 primes
Probability = (4/9)*(4/9)*(4/9)*(4/9)

Probability of at least 2 primes = 1 - (Probability of 0 prime + Probability of 1 prime)
Probability of at least 2 primes = Probability of 2 primes + Probability of 3 primes + Probability of 4 primes

The calculations are painful so let's leave it here. As I said, their intention was probability of prime = 1/2, probability of composite = 1/2 which makes the calculations simple.

The 4 positions are different but you can have the same prime on one or more positions. When you say 4/9, you are including all cases (2, 3, 5, 7) so you don't need to account for them separately. When you say, (4/9)*(4/9)*(4/9)*(4/9), you are including all cases e.g. (2222, 2353, 3577, 2357 etc). All you need to do it separate out the primes and the non primes. That you do by arranging primes and non primes as NNPP or NPNP or PPNN etc (as we did above)
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Just a small confusion
2,3,5,7 are primes , and 1,4,6,8,9 non primes ,just because there are 3 one's in the last three places it doesn't mean that 1 cannot be at any other place, question says boy remembers last three places having one's and not that there are only 3 ones in the 7 digit number
the phone number could be {1 2 3 6 1 1 1 } this has two primes and 5 non primes and 4 one's
or { 8 2 2 6 1 1 1 } this has 2 primes and all the primes are same

first going the long way p( 2 primes ) +p( 3 primes )+p(4 primes )

\(\frac{4}{9} * \frac{4}{9} * \frac{5}{9} *\frac{5}{9} *1*1*1* \frac{4!}{2!2!}\) ( exactly two primes )
Multiplying by \(\frac{4!}{2!2!}\) as we can permutate only the first 4 digits , the last are fixed (1,1,1)

\(\frac{4}{9} * \frac{4}{9}* \frac{4}{9}*\frac{5}{9}*1*1*1 *\frac{4!}{3!}\) ( exactly 3 primes )( 3 of a kind )

\(\frac{4}{9} *\frac{4}{9} *\frac{4}{9} *\frac{4}{9} * 1,1,1\) ( exactly 4 primes )

Now for the case 2 primes, both the primes could be same or different then how does the notation \(\frac{4!}{2!2!}\) change , or does it remain the same ?

Similarly for the case of 3 primes the primes could be 2,2,2 all same or 2,3,5 all different then how does the notation \(\frac{4!}{3! }\) change or does it remain the same?

In this question we are taking primes as one kind and non primes as other kind , so it doesn't matter if the primes are all same or all different ? Is this statement correct?

1)Please could you show how to do this sum individual probability way as I have tried above ?

2)Also please consider the fact that 1 may have to be included as a non prime as the question does not explicitly state that there are only 3 ones in the phone number , he only remembers that the last three are ones.

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Thank you karishma, since there was no response for a while I thought my query was unreasonable, good to see at least some people appreciated what I wrote . Highly appreciate your response.

Great so now I know probability of one prime 4/9* 5/9 * 5/9 *5/9 * 4= 2000/6561

PROBABILITY of 0 prime's = 5/9*5/9*5/9*5/9= 625/6561

probability of at least 2 primes = 1- ( 2000/6561 + 625/6561)
1-(2625/6561) = 3936/6561 = .59

now lets check by doing long way

2 primes

4/9 * 4/9 * 5/9*5/9* 4!/2!2! = 2400/6561 ( PPNN : two primes could be same ,2 , 2 or different 2, 3 doesn't matter we can think of this as,
two of a kind and two of another kind ,hence 4!/2!2!)

3 primes

4/9 *4/9 *4/9*5/9*4 = 1280/6561 (4!/3! = 4) Three of one kind again PPPN.

4 primes

4/9*4/9*4/9*4/9 = 256/6561 all of the same kind , hence only one way to select them. PPPP


total 3936/6561 = .59 Bingo!

hence we can see both ways we are getting the same result.

Karishma if there is any error in my understanding, please do point out, I have considered primes as one kind and non primes as another kind instead of the position logic which you have mentioned.
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Everything seems to be in order. It doesn't matter whether you choose to think in terms of position or arrangement of Ps and Ns. The result would be the same.
When I need all arrangements of PNNN, I can write it as 4!/3! (= 4) or I can say that I will select one place for the P in 4 different ways.
Either ways, we are arranging a P and 3 Ns.
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Nice problem but I think we should change the language just to clarify that 1 can't be used again even if it appears on te last 4 digits.

Thanks mods

Cheers!
J

I agree that we should change the language so that the question will be clearer than now. At first I also assume that 1 appears once within 3 last places.

Hi! I have a question here. What does is wrong if we do this in this way?
Outcome wanted/Total outcome. This means:

Outcome wanted: 4*4*8*8*1*1*1. I have in the first and in the second digit, 4 possibles numbers (just primes). For the third and fourth digit are 8 possible numbers (2,3,4,5,6,7,8 and 9).
Total outcome: 8*8*8*8*1*1*1

Please help!

Hey gbascurs,
The question ask "What is the probability that the phone number contains at least two prime digits?". At least mean: 2, 3, or 4 primes digits appears in the phone number. You only include one case. Moreover, the two primes can be in the third and fourth digit, not just in the first and second digit.

Hope it clears.

Hi LucyDang,

Why I am trying to do now is:

Probability of 2 primes = 4*4*8*8*1*1*1 /8*8*8*8*1*1*1 +
Probability of 3 primes = 4*4*4*8*1*1*1 /8*8*8*8*1*1*1 +
Probability of 4 primes = 4*4*4*4*1*1*1 /8*8*8*8*1*1*1
Probability = 7/16

I cannot see where is the mistake. Thanks!!

Hey gbascurs,

There are overlap/incorrect calculation here:
Probability of 2 primes = 4*4*8*8*1*1*1 /8*8*8*8*1*1*1 --> 4*4*8*8*1*1*1: there are 4 primes each placed in 3rd and 4th digit
Probability of 3 primes = 4*4*4*8*1*1*1 /8*8*8*8*1*1*1 --> 4*4*4*8*1*1*1: there are 4 primes placed in 4th digit.
You have to calculate the case in which same two primes appear, then you need to subtract the primes from the total numbers. It takes lots of time to do so, so please use the P(desire) = 1-P(opposite) to get the correct answer under time pressure.

actually, 1 is neither prime nor composite number, so, the single digit prime numbers are 2, 3, 5, 7. (2 is the only even prime number).

XXXX(1}{1}{1}
X can be a prime number or non prime number
Prime number:2,3,5,7
Non prime-4,6,8,9(1 is already taken)
Probability(X is a Prime number)=4/8=1/2
Probability(X is a non-prime number)=4/8=1/2

So,Atleast 2 prime numbers=P P NP NP=4!/2!2!*1/2*1/2*1/2*1/2=6/16
Three prime numbers and one Non prime=P P P NP=4!/3!*1//2^4=4/16
All prime=PPPP=1/16
Total=11/16.

Remember that its a telelphone number,so different arrangements give you different number-same goes for codes,passwords,words..

Can you please explain me why you have multiplied with 4!/2!2!? We do have 2 primes but they could be different, right?

So, how are we sure they are same?
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Unless mentioned in the question that repetition is not allowed or different digits have to be used,we should not assume anything on our own.
Then,it is clearly mentioned in the question that last three digits are 1,so you have to fix the number 1 and 0 is not used.So,we are left with 8 digits.
There can be numbers such as 2244 or say 3366 etc of the format P P NP NP.It's not necessary that they have to be different.It's nowhere mentioned in the question.So,if you have 2244-how many ways can you arrange it-4!/2!2!=6 ways (2 is repeated 2 times and 4 is repeated 2 times),so divide it by the number of possible arrangement of the number to avoid duplicate numbers.For example 2244 can be written as 2424 2442 4242 4422 4224 2244- 6 possible ways.

4!/2!2! means the arrangement of P P NP NP in which 2 objects are the same(prime numbers) and 2 Non prime numbers are the same.If you wouldn't divide it by 2! and 2!,there will be duplicate arrangements.
Its the same as arrangement of say word BANANA=you have 6 letters-so they can be arranged in 6! ways/3!(letter A)2!(letter N)

ANSWER IS B

Prime is P
no Prime is X
P(prime) = P(no prime) = 4/8 =0.5
PPPP111___________4!/4! x (0.5)^4
PPPX111___________4!/3! x (0.5)^4
PPXX111___________4!/2!2! x (0.5)^4
Summation = (0.5)^4 { 1+4+6}= 11/16 = Option B

Total number of ways in which remaining 4 places can be filled from 8 digits = 8 x 8 x 8 x 8
2 Prime (from 4 prime nos), 2 non prime (from 4 non prime nos) = 4^4 x 4C2
3 Prime, 1 non prime = 4^4 x 4C3
4 Prime, 0 non prime = 4^4 x 4C4
Probability = 4^4(4C2+4C3+4C4)/8^4 = 11/16

We know that the single-digit numbers that are primes are 2, 3, 5, and 7.

Since the last three digits are all 1’s, we only have to deal with the first 4 digits, in which none of them is 0 (and we are also going to assume that none of the first 4 digits is 1).

Thus, the probability that the first 4 digits are all primes (where each digit can be 2 to 9, inclusive) is:

4/8 x 4/8 x 4/8 x 4/8 = (1/2)^4 = 1/16

The probability that exactly 3 of the first 4 digits are primes is:

(4/8 x 4/8 x 4/8 x 4/8) x 4C3 = (1/2)^4 x 4 = 4/16

Notice that the last factor 4C3 is the number of ways one can arrange 3 prime digits in 4 spots. The probability that exactly 2 of the first 4 digits are primes is:

(4/8 x 4/8 x 4/8 x 4/8) x 4C2 = (1/2)^4 x 6 = 6/16

Thus, the probability that the phone number contains at least two prime digits is:

1/16 + 4/16 + 6/16 = 11/16

Answer: B
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1 can't be in any of the 4 remaining spots. So we have: 2,3,5,7, 4,6,8 and 9.
total number of arrangements of 8 numbers in 4 slots is 8*8*8*8=8^4
for two primes with the possibility of repeating numbers is 4*4*4*4 (first two fours are for the primes and the second one is for the non primes)
for three primes and four primes we have 4*4*4*4. However, the three can be arranged as follows:
PP(NP)(NP): 4!/(2!2!)=6
PPP(NP): 4!/3!=4
PPPP: 4!/4!=1

4^4/8^8=4^4/(2^4*4^4)=1/16
Adding 6,4 and 1 and multiplying by their probability we get 11/16.