Question regarding binomial random variable and distribution

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The discussion clarifies the concept of binomial random variables in probability, specifically focusing on the outcomes of a binomial experiment with three trials (n = 3). It establishes that each trial has two possible outcomes (S or F), leading to a total of 8 possible outcomes calculated using the formula 2^n, where n represents the number of trials. The conversation also distinguishes between scenarios where order matters and where it does not, referencing the binomial coefficient for cases where order is irrelevant.

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Hi,

just started learning probability & need some help in understanding...

"The binomial random variable X associated with a binomial experiment consisting of n trials is defined as

X = the number of S's among the n trials.

Suppose, for example, that n = 3. Then there are 8 possible outcomes for the experiment:

SSS SSF SFS SFF FSS FSF FFS FFF"

Why is it that there will be 8 possible outcomes?
 
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You have three trials--each trial has two possible results: S or F. Computing the total number of outcomes of the three trials is a matter of just adding together all of the separate results.

First trial: 2 results, Second trial: 2 results, Third trial: 2 results

2*2*2 = 23 = 8 outcomes

The general formula is: Number of outcomes = pn, where p is the number of possible results of each trial, and n is the number of trials.

If order doesn't matter, then in your example SSF and FSS would be considered the same outcome since the same number of S and F occur in both. The formula for this is different. Here's a link for more, I'm late for class...

http://en.wikipedia.org/wiki/Binomial_coefficient
http://mathworld.wolfram.com/Combination.html
 
Thanks very much for the explanation. I understand why there 8 possible outcomes now. thanks again!
 

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