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Part III: Distributions and the Central Limit Theorem

2. Does each trial have only two possible outcomes — success or failure?

The outcome of each flip is either heads or tails, and you’re interested in
counting the number of heads. That means success = heads, and failure
= tails. Condition 2 is met.

3. Is the probability of success the same for each trial?
Because the coin is fair, the probability of success (getting a head) is

p = ⁄2 for each trial. You also know that 1 – ⁄2 = ⁄2 is the probability of fail-
1
ure (getting a tail) on each trial. Condition 3 is met.
4. Are the trials independent?

You assume the coin is being flipped the same way each time, which

means the outcome of one flip doesn’t affect the outcome of subsequent
flips. Condition 4 is met.
Because the random variable X (the number of successes [heads] that occur
in 10 trials [flips]) meets all four conditions, you conclude it has a binomial
1
distribution with n = 10 and p = ⁄2. 1 1
But not every situation that appears binomial actually is. Read on to see
some examples of what I mean.
No fixed number of trials
Suppose that you’re going to flip a fair coin until you get four heads and you’ll
count how many flips it takes to get there; in this case X = number of flips.
This certainly sounds like a binomial situation: Condition 2 is met because
you have success (heads) and failure (tails) on each flip; condition 3 is met
with the probability of success (heads) being the same (0.5) on each flip; and
the flips are independent, so condition 4 is met.
However, notice that X isn’t counting the number of heads, it counts the
number of trials needed to get 4 heads. The number of successes (X) is fixed
rather than the number of trials (n). Condition 1 is not met, so X does not
have a binomial distribution in this case.
More than success or failure
Some situations involve more than two possible outcomes, yet they can
appear to be binomial. For example, suppose you roll a fair die 10 times and
let X be the outcome of each roll (1, 2, 3, . . . , 6). You have a series of n =
10 trials, they are independent, and the probability of each outcome is the
same for each roll. However, on each roll you’re recording the outcome on a
six-sided die, a number from 1 to 6. This is not a success/failure situation, so
condition 2 is not met.

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