Page 157 - Statistics for Dummies

Chapter 8: Random Variables and the Binomial Distribution

2. Find the column that represents your particular value of p (or the one
closest to it, if appropriate).
3. Find the row that represents the number of successes (x) you are

interested in.

4. Intersect the row and column from Steps 2 and 3. This gives you the
probability for x successes, written as p(x).
For the traffic light example from “Finding Binomial Probabilities Using a
Formula,” you can use the binomial table (Table A-3 in the appendix) to verify
the results found by the binomial formula shown back in Table 8-2. Go to the
mini-table where n =3 and look in the column where p = 0.30. You see four
probabilities listed for this mini-table: 0.343, 0.441, 0.189, and 0.027; these are
the probabilities for X = 0, 1, 2, and 3 red lights, respectively, matching those
from Table 8-2.
Finding probabilities for X greater-than, 141
less-than, or between two values
The binomial table (Table A-3 in the appendix) shows probabilities for X
being equal to any value from 0 to n, for a variety of ps. To find probabilities
for X being less-than, greater-than, or between two values, just find the cor-
responding values in the table and add their probabilities. For the traffic light
example, you count the number of times (X) that you hit a red light (out of
3 possible lights). Each light has a 0.30 chance of being red, so you have a
binomial distribution with n = 3 and p = 0.30. If you want the probability that
you hit more than one red light, you find p(x > 1) by adding p(2) + p(3) from
Table A-3 to get 0.189 + 0.027 = 0.216.
The probability that you hit between 1 and 3 (inclusive) red lights is
p(1 ≤ x ≤ 3) = 0.441 + 0.189 + 0.027 = 0.657.
You have to distinguish between a greater-than (>) and a greater-than-or-equal-
to (≥) probability when working with discrete random variables. Repackaging
the previous two examples, you see p(x > 1) = 0.216 but p(x ≥ 1) = 0.657. This is
a non-issue for continuous random variables (see Chapter 9).
Other phrases to remember: at least means that number or higher, and at most
means that number or lower. For example, the probability that X is at least 2 is
p(x ≥ 2); the probability that X is at most 2 is p(x ≤ 2).

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