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Here is the question, from this video

In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?

So if the chicks are equally likely to peck left or right, the answer is 25.

If we consider a single chick, the chance it is pecked is $3/4$ and the chance it isn't is $1/4$.

Let the random variable be

$$Z = \begin{cases} 0, & \text{if chick is pecked} \\ 1, & \text{if chick is not pecked} \end{cases}$$

and so for each chick $$E[Z_i] = 0 * P(0) + 1 * P(1)$$ $$= 1/4$$

And by the linearity of expectation, we get 25.

$E[Z_i]$ is basically the probability that chick $i$ is not pecked. By my understanding, $Z$ can map any value to any outcome, it's our choice.

So what if we reversed, and had

$$Z = \begin{cases} 1, & \text{if chick is pecked} \\ 0, & \text{if chick is not pecked} \end{cases}$$

Then $E[Z_i] = 3/ 4$, and our answer would be 75.

Or what if we had

$$Z = \begin{cases} 331, & \text{if chick is pecked} \\ 798, & \text{if chick is not pecked} \end{cases}$$

then $E[Z_i] = 1791/ 4$, and the sum of the expected values would be 44,775. How do we choose the correct values for the random variables, or am I interpreting the results wrong?

I really don't understand the point of random variables. We could have easily deduced the answer 25, by saying that there is a 25% chance a chick is not pecked, and so for 100 chicks, 25 of them will be not be pecked. We really don't even need to talk about expected value either.

Rockstar5645
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  • I think it is more about how you interpret what $Z$ represents. In the cases where set the $Z$ values to 0/1 and 1/0, it just so happens that the expected value ends up equalling the probability that a chick is or is not pecked. When you assign the values 331/798 we need to interpret $Z$ in a different way, it could be for example, the value of a chick to the farmer and $E[Z_i]$ ends up being the expected value of the $i^{th}$ chick. – Paul Aljabar Jun 08 '17 at 06:45
  • I guess I just don't understand the whole point of random variables, let me extend my question. – Rockstar5645 Jun 08 '17 at 06:47
  • If Z is the value of a chick to a farmer, what would 44,775 represent? – Rockstar5645 Jun 08 '17 at 06:53
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    The random variable you're really interested in is the number of unpecked chicks. Call this variable $Y$. It can take values between $0$ and $50$. We have $Y = Z_1 + Z_2 + \dots + Z_{100}$, where the variables $Z_i$ are defined as you defined them the first time. This would not be true if you defined $Z_i$ the other ways. The way the concept of a random variable helped in this instance is that you were able to analyze a complicated variable, $Y$, in terms of much simpler variables. – user49640 Jun 08 '17 at 07:01
  • Okay, $Y=Z_1+Z_2+⋯+Z_{100}$ does give me the answer, and if I defined them the second way, we get the expected number of chicks that were pecked. It makes intuitive sense, but again, for the answer, $Z_i$ is just the probability of a chick not getting pecked. What's the point of random variables here? – Rockstar5645 Jun 08 '17 at 07:43
  • When you take the expectation, you're always taking the expectation of something. That something is always a random variable. Here, $Y$ is actually quite a complicated random variable, even though it is the sum of simple ones. – user49640 Jun 08 '17 at 07:44
  • Anyway, if that seems too easy, here's another question. What is the expected value of the square of the number of unpecked chicks? The answer is 631.25. You get the answer by expanding the expression for $Y^2$, and evaluating the expected value of $Z_i Z_j$ in different cases. The cases $j = i$ and $j = i \pm 2$ (cyclically modulo 100) need to be treated separately. – user49640 Jun 08 '17 at 08:06
  • "expected value of the square of the number of unpecked chicks", I'm having trouble understanding what this means. Isn't it just the square of the number of unpecked chicks? 625? – Rockstar5645 Jun 08 '17 at 08:19
  • Do you think the square of the number of unpecked chicks is always 625? Never 576 and never 676? – David K Jun 08 '17 at 13:06
  • Oh yea, I guess it could be those values too. – Rockstar5645 Jun 08 '17 at 15:56
  • Related: https://math.stackexchange.com/questions/2282622/expected-number-of-unpecked-chicks-nyt-article – Henry Jun 11 '17 at 21:35

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