A license plate in the country Kerrania consists of four digits follow

Question

A license plate in the country Kerrania consists of four digits followed by two letters. The letters A, B, and C are used only by government vehicles while the letters D through Z are used by non-government vehicles. Kerrania's intelligence agency has recently captured a message from the country Gonzalia indicating that an electronic transmitter has been installed in a Kerrania government vehicle with a license plate starting with 79. If it takes the police 10 minutes to inspect each vehicle, what is the probability that the police will find the transmitter within three hours?

(A) 18/79
(B) 1/6
(C) 1/25
(D) 1/50
(E) 1/900

Show Answer

Official Answer

D

(A) 18/79 
(B) 1/6 
(C) 1/25 
(D) 1/50 
(E) 1/900

UrsTruly · Accepted Answer

If it takes 10 minutes to inspect one vehicle, the # of vehicles that can be inspected in 3 hours (180 minutes) = 180/10 = 18. Hence, for calculating the probability that the police will find the transmitter within three hours, the favorable cases = 18. Now, we need to figure out the total # of cases.

The total # of cases = Total # of such cars possible. The details given about the car is that it starts with 79, which leaves 2 more digits, both of which can be filled by all 10 numbers (0-9). In addition, we have 3 letters, each of which can be filled by any from the set {A,B,C}. Hence the total # of such cars possible = 10*10*3*3 = 900

So, the probability that the police will find the transmitter within three hours = 18/900 = 1/50. Option D

dangeyg · Answer

the language four digits followed by two letters, it sounded like total 6 alphanumerics. can you please comment about this.

EMPOWERgmatRichC · Answer

Hi dangeyg,

This question involves permutations AND probability; these questions also tend to be very specific about what they ask for, so you have to pay careful attention to the details and take the proper notes.

We're given a variety of Facts about license plates in Kerrania:
1) Plates are always 6 characters - 4 digits followed by 2 letters (in THAT order)
2) On GOVERNMENT plates, the ONLY 3 letters that are used are A, B and C.
3) A specific GOVERNMENT plate that BEGINS with 79 _ _ _ _ is being sought.

To start this question off, we need to figure out how many DIFFERENT plates fit these "restrictions"

Since the first 4 spaces on a plate are DIGITS, the 'third' and 'fourth' spots can be anything from 0 through 9, inclusive (this means there are 10 options for each of those two spots). Next, since we're dealing with a GOVERNMENT vehicle, the 'fifth' and 'sixth' spots can only be the letters A, B and C (so there are 3 options for each of those two spots).

(10)(10)(3)(3) = 900 possible government plates begin with 79 _ _ _ _. UrsTruly explains the rest of the math correctly, so I won't rehash any of that here.

GMAT assassins aren't born, they're made,
Rich

mvictor · Answer

tough and tricky one
we know that the plates start with 79
the plate consist of 4 digits and 2 letters
we have these possibilities
1st digit known, so only 1 possibility
2nd digit known, so only 1 possibility
3rd digit unknown, so 10 possibilities
4th digit unknown, so 10 possibilities

we have 100 possibilities by digits

now we know that only letters A, B, and C are used for government's vehicles, thus, we can establish these possibilities
1st letter, A, B, or C, which gives us 3 possibilities
2nd letter, A, B, or C, again 3 possibilities.
checking how many combinations we can have, 3x3 = 9, so we have 9 possibilities.

taking all together, by number of digits and by letters, we have 900 plates.

to check each car, it takes 10 minutes, so for all plates, we need 9000 minutes.
3 hours, that's 180 minutes.

the probability of catching the suspect is thus 180/9000, simplify 18/900 = 2/100 = 1/50
answer choice D.

anairamitch1804 · Answer

Since the first two digits of the license plate are known and there are 10 possibilities for each of the remaining two digits (each can be any digit from 0 to 9), the total number of combinations for digits on the license plate will equal 10 ×10 = 100. 
Because there are only 3 letters that can be used for government license plates (A, B, or C), there are a total of nine two-letter combinations that could be on the license plate (3 possibilities for first letter × 3 possibilities for the second letter).

Given that we have 100 possible digit combinations and 9 possible letter combinations, the total number of vehicles to be inspected will equal 100 × 9 = 900. 
Since it takes 10 minutes to inspect one vehicle, the police will have time to inspect 18 vehicles in three hours (3 hours = 180 minutes). Thus, the probability of locating the transmitter within the allotted time is 18/900 = 1/50. The correct answer is D.

falewman · Answer

Since the first two digits are already determined, the inspectors will only have to sift through two more digits and two letters.

The Police could search one car in 10 mins. Altogether they can search 6X3 =  18 cars  in 3 hours.

The 
For the family of plates starting with 79, the plate has the format -- 79_ _ _ _
For the third digit and the fourth digit the possible options are numbers from 0 to 9
For the last two digits, the options are A, B, C.
Hence the possible plates are 10 X 10 X 3 X 3 =  900

The probability of finding the transmitter in 3 hours = 18/900 = 1/50

My Answer is "D"

Nunuboy1994 · Answer

This problem contains superfluous information; well more so, not all the information given in the problem will be used to calculate our answer...let's take it step by step
So this license plate will have 6 characters: 4 randomized digits (these digits DO NOT have to be distinct- the problem does explicitly state this so it is assumed that digits can repeat such as 7999AAA or 7979BBB). Kerrania's Intelligence Agency has already identified the first two digits for this unique six character license place so we just multiply the possibilities for the last two numerical digits times the two possibilities for the last two letters which must derive from the set of Government letters only. Hence,

10 (numbers 0-9) * 10 (numbers 0-9) * 3 (letters A-C) * 3 (letters A-C) = 900 cars that start with 79 and use strictly Government letters

Now, the problem states it takes 10 minutes to search for each car...so how many cars can Kerrania's intelligence agency search through in 3 hours? Well, there are 60 minutes in 1 hour- so just multiply 60 minutes by 3 for a total of 180 minutes-so if Kerrania's intelligence agency can search through 1 car in 10 minutes then in 180 minutes they can search for 18 cars. Essentially what this question is asking is the total number of cars that can be found in an hour divided by the total number of cars in order to calculate the probability of finding a particular car within that group of 900 cars that contains the transmitter- it can be a very confounding problem so patience is necessary

P(cars found in 3 hours)/ P(total possibility of cars starting with 79 and using government letters)

18 cars / 900 cars = 1/50

ShashankDave · Answer

Hi, this question really gave me some problems,but I could finally do it. My question from you is derived from the following that you have written:-

Since it takes 10 minutes to inspect one vehicle, the police will have time to inspect 18 vehicles in three hours (3 hours = 180 minutes). Thus, the probability of locating the transmitter within the allotted time is 18/900 = 1/50.

I agree that in 3 hours, the number of vehicles that can be tested is 18, and we have 900 vehicles in total that may be tested.

What we have to find, is the probability that we find our transmitter vehicle in no more than 18 attempts. That would translate into - the first vehicle is our vehicle or the first is some other vehicle and then our vehicle or .....

What you have done is simply divided The cars we can check by The total number of cars that are there.

I cannot understand the connection, how does this give us the answer? please help. I am trying to actually fully understand the concepts of probability, this would really help fetch a higher score. Thanks a lot

TimeTraveller · Answer

Probability that a certain event will happen = No. of favourable events/Sample space.

Favourable events in this question = fishing out the vehicle with the said license plate in 180 mins.
Sample space = total time it would require to go through all the vehicles matching the said criteria (79-xx-yy).

We have 900 vehicles to inspect. It takes 10 minutes to inspect one vehicle. So, the time it will take to inspect 900 vehicles will be 9000 minutes.

We have 180 minutes to do the search. So, the probability of succeeding = 180/9000 = 1/50. Hope its clear.

Contropositive · Answer

Bunuel   KarishmaB   chetan2u 
I get that, here, Total number of cases = 900
but how is that favorable cases are ''18''?

Its possible that none of those 18 cars checked had transmitter.

Moreover, the total probability to find the correct car will be 1/900, so the segmented probability(finding the car in 3hrs) should be less than 1/900, right?

chetan2u · Answer

The probability that it is first car is 1/900. Probability that it is the second car is again 1/900 and so on as the probability of any car would be same in an unbiased situation.

So for first 18 cars, the combined probability will be sum of each probability = (1/900)+(1/900)+…..18 times = 18*1/900 = 1/50

Posted from my mobile device

A_Nishith · Answer

Since the first two digits are already determined, the inspectors will only have to sift through two more digits and two letters.  79_ _ _ _

For the Number plates starting with 79, the plate has the format -- 79_ _ _ _
For the third digit and the fourth digit the possible options are numbers from 0 to 9.
For the last two digits, as it is given that Govt. Vehicle is searched, so the options are A, B, or C.

3rd digit unknown, so 10 possibilities
4th digit unknown, so 10 possibilities
=> we have 10x10=100 possibilities by digits

now we know that only letters A, B, and C are used for government's vehicles, thus, we can establish these possibilities
1st letter, A, B, or C, which gives us 3 possibilities
2nd letter, A, B, or C, again 3 possibilities. ( As there is no restriction for repeating the alphabets)
=> we have 3x3=9 possibilities by alphabets

Hence the possible plates are 10 X 10 X 3 X 3 =  900  
The Police could search one car in 10 mins. Altogether they can search 6X3 =  18 cars  in 3 hours.
The probability of finding the transmitter in 3 hours =  18/900 = 1/50

My Answer is "D"

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A license plate in the country Kerrania consists of four digits follow

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