Consider a series of 300 numbers in which the first number is 3 and th

Question

Consider a sequence of 300 numbers in which the first number is 3 and the second number is 5, and each succeeding number is equal to the sum of the previous two numbers in the series. If a number is randomly selected from the series, what is the probability that the number selected will be odd?

A. 1/3
B. 1/2
C. 2/3
D. 3/4
E. 4/5

A. 1/3
B. 1/2
C. 2/3
D. 3/4
E. 4/5

twobagels · Answer

I calculated out the first 6 terms: 3, 5, 8, 13, 21, 34.

I assume the pattern simply repeats 2 odd to 1 even, hence a probability of 2/3 of picking an odd number at random.

ET10 · Answer

I approached this doing a micro-sample to try to find a pattern.

First number: 3 = odd
Second number: 5 = odd
Third number: 8 = even
Fourth number: 13 = odd
Fifth number: 21 = odd
Sixth number: 34 = even

From here, I see a pattern that every third number is even. So in a sequence of 300, 1/3 will be even and 2/3 will be odd.

IMO C

stne · Answer

Dear  IanStewart  Sir, 
How do I know what is the correct size of mini sample I should take?
I tried with a sample size of 10 and got a different value

3, 5, 8, 13, 21, 34, 55, odd, even, odd. 
Here we have a total of 10 numbers out of which 7 are odd and 3 are even.
if 10 numbers have 7 odd then 300 numbers will have 210 odd 
Hence probability is  \frac{210}{300} = \frac{7}{10}

What am I missing?

IanStewart · Answer

If you just looked at the first few terms of this sequence, without understanding the pattern of evens and odds in the sequence, it would really just be good luck if you got the right answer. Here, we add the two previous terms to find the next term. Our sequence starts with two odd numbers, so using odd/even arithmetic rules, the sequence will look like this:

odd, odd, even, odd, odd, even, odd, odd, even, ...

and because odd and even rules never change, this pattern will continue indefinitely. We can see that in each block of three terms, we have two odd values, so 2/3 is the answer (here I'm using the fact that 300 is divisible by 3 -- if we had to look at, say, 100 terms, we'd know 66 of the first 99 are odd, and the 100th is odd too, so then the answer would be 67/100).

The wording here should be changed, incidentally - a "series" in math is not the same thing as a "sequence", and this question is using the word "series" incorrectly.

stne · Answer

Sir, just as 3 divides 300 so does 100 and 10 along with many others. So how do I know whether I should choose 3, 10 or 100 as the sample size.
Shouldn't all the sample size give the same probability?

IanStewart · Answer

No, certainly not. It's usually just good luck if that happens. Just take any question you know the answer to: what is the probability one of the first 100 positive integers is odd? The answer is clearly 1/2. But if you look at just the first three integers: 1, 2, 3, it is not true that 1/2 of them are odd.

In this particular question, the first three numbers in the sequence are odd, odd, even, then the next three are odd, odd, even, and so on. So if we look at the first 3 terms, or 6 terms, or 9 terms, or 300 terms, exactly 2/3 of them will be odd. It's because the pattern repeats in a set of three terms that it's important that 300 is divisible by 3.

stne · Answer

Thank you Sir, got it. 
I cannot take 10 as the same size because the pattern is not consistent with a sample size of 10

o o e oo e oo e o - Here we have 7 odd and 3 even
o e o o e o o e o o- Here also we have 7 odd and 3 even 
e o o e o o e o o e- But here we have 6 odd and 4 even

Hence we cannot choose a sample size that has an inconsistent pattern.

Sample size of 3 will always have 2 odd and one even hence we can safely choose 3 as the sample size. Is this the crux of the matter?

bumpbot · Answer

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Consider a series of 300 numbers in which the first number is 3 and th

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Consider a series of 300 numbers in which the first number is 3 and th

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