This is a summary for the subject matter of lecture 8 of Descriptive Statistics in the pre-master in Orthopedagogy at the University of Amsterdam. It covers chapters 6.3 and 7.1 of Algresti & Franklin (Statistics).
6.3. Probabilities when each observation has two
possible outcomes
The binomial distribution: probabilities for counts with binary data
- In many applications, each observation is binary: it has one of two possible outcomes
- With a sample, we summarize such variables by counting the number or the proportion of
cases with an outcome of interest.
- Under certain conditions, a random variable X that counts the number of observations of a
particular type has binominal (probability) distribution. The binominal distribution
specifies probabilities for these possible values of X when the following conditions hold.
- Conditions for binominal distribution:
o Each of n trials has two outcomes. The outcome of interest is called a success and
the other outcome is called a failure.
o Each trial has the same probability of success. This is denoted by p, so the
probability of a success is p and the probability of a failure is 1 – p
o The n trials are independent. That is, the result for one trial doesn’t depend on the
results of other trials.
- The binominal random variable X is the number of successes in the n trials
- E.g. coin flip
Probabilities for a binomial distribution
- Denote the probability of success on a trial by p. for n independent trials, the probability of x
n! x n−x
successes equals P ( x )= p (1− p) x=0 , 1 ,2 … , n
x ! ( n−x ) !
- The symbol n! is called n factorial. It represents n !=1× 2× 3 ×… ×n , the product of all
integers from 1 to n. Also, 0! Is defined to be 1. For given values for p and n, you can find the
probabilities of the possible outcomes by substituting values for x into the binomial formula.
- The factorial term tells us the number of possible outcomes that have x successes.
- The term p x (1− p)n−x with the exponents gives the probability for each such sequence.
- The binominal distribution is only symmetric when p=0.5, otherwise it has a skewed
appearance. The degree of skew increases as p gets closer to 0 or 1.
Check to see if binomial conditions apply
- Before you use the binomial distributions, check that its three conditions apply.
o (1) binary data (success or failure), (2) same probability of success per trial and (3) a
fixed number n of independent trials.
- To judge this, ask yourself whether the observations resemble coin flipping.
Population and sample sizes to use the binomial
- For sampling n separate subjects from a population (that is, sampling without replacement),
the exact probability distribution of the number of successes is too complex to discuss in this
text, but the binomial distribution approximates it well when n is less than 10% of the
population size. In practice, sample sizes are usually small compared to population sizes, and
this guideline is satisfied.
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