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A certain machine produces toy cars in an infinitely repeati

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A certain machine produces toy cars in an infinitely repeati [#permalink] New post 21 Feb 2018, 07:14
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A certain machine produces toy cars in an infinitely repeating cycle of blue, red, green, yellow and black. If 6 consecutively produced cars are selected at random, what is the probability that 2 of the cars selected are red?

A. \(\frac{1}{6}\)

B. \(\frac{1}{5}\)

C. \(\frac{1}{3}\)

D. \(\frac{2}{5}\)

E. \(\frac{1}{2}\)

[Reveal] Spoiler: OE
We need to find possible and desired outcomes. What are the possible outcomes here? Well, we're going to be randomly selecting 6 consecutively produced cars and we want to know what the probability is that 2 of those cars will be red. How many different groups of 6 consecutive cars are there? Well, we've got 5 colors, so any one of them could be the first in the string of 6 that we choose. That gives us 5 different strings of 6. So 5 is the number of possible outcomes here. What about desired outcomes?



We need to know how many of those 5 strings have 2 red cars. What does getting two red cars depend on? It depends on the color of the first car of the string of 6. For example, if blue is the first car of the string, then you'd get blue, red, green, yellow, black and then blue again.



So, the only way to get 2 red cars is if the first car of the string is red, because then we'd have red, green, yellow, black, blue and then red again as the sixth car. So 1 out of the 5 strings will contain 2 red cars, so that's a probability of 1/5
[Reveal] Spoiler: OA

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Re: A certain machine produces toy cars in an infinitely repeati [#permalink] New post 21 Feb 2018, 10:34
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This problem is a disguised version of the following question: if you have pick any 6 consecutive integers, what are the odds that two of them are divisible by 5? It's good to know that every fifth integer is divisible by 5, every 7th divisible by 7, etc.

Why is this so? If we're trying to find the odds that two are divisible by 5, we could select some groups of 6 integers and see what happens:
123456
234567
345678
456789
5678910

Notice every set of integers has one 5 in it, but in the last set, if we start with a 5, we'll end on a 10, giving us two integers divisible by 5. If we scoot one over again we'll lose the 5 and have a 10 in its place for the next 4 sets. Long story short, every fifth group has two numbers divisible by 5. Let's apply it to this problem:

BRGYBB
RGYBBR
GYBBRG
YBBRGY
BBRGYB

Since these are the only 5 ways you can select 6 cars in this order, and since only one has a red car twice, the answer is 1/5, or B.
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Re: A certain machine produces toy cars in an infinitely repeati   [#permalink] 21 Feb 2018, 10:34
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