STATISTICAL TOPICS AND ANALYSIS IN RISK ASSESSMENT - Risk Management Applications in Pharmaceutical and Biopharmaceutical Manufacturing - page 75

The probability of flipping a nickel and getting a head is in theory one-half,
0.5, or 50%.

P (Head) = 0.5

In actual practice, it could be slightly more or less. The old buffalo nickel was
heavy on the side with the buffalo. It is possible that it could be shown to have
an effect.
The numerator and denominator in probability calculation need to be carefully
defined. The letters a, e, i, o, u, and y are the usual vowels. Thus, the probability
of picking a vowel at random from a page of text is 6 out of 26 or 23%. In some
cases, however, y is used as a consonant. Thus, there is some uncertainty in the
numerator and the final result. Careful assumptions, operational definitions, and
specific rules will clarify these situations.
What is the probability of tossing a nickel twice and getting two heads? The
probability is assumed to be one-half for the first toss and one-half for the second
toss. Thus, 0.5 AND 0.5, where AND means to multiply. The result is one out
of four.

P (Head and Head) = P (Head) ∗ P (Head) = 0.5 ∗ 0.5 = 0.25 or 25%.

We multiply when we see “AND.”
This assumes that the two events are independent of each other. That is, they
do not influence each other. If the events are not independent, the rule or formula
is shown here.
What is the probability of getting a head or a tail on one toss of a dime? The
probability is assumed to be one-half for a head or one-half for a tail. Thus, 0.5
OR 0.5, where OR means to add.

P (Head or Tail) = P (Head) + P (Tail) = 0.5 + 0.5 = 1.0 or 100%

This holds as long as there is no commonality of the two events. We add when
we see “OR.”

4.6 CAUTIONS

The cumulative effect of these rules can be impressive. For example, conducting
multiple t -tests with the same alpha value can result in a very high probability
of at least one failure just by random chance. Assume an assay validation report
has 47 t -tests, each with a typical alpha, or α value of 0.05. The probability of
at least one or more failures is found by the formula 1 − (1 − α ) n .
So,

n

P = 1 − ( 1 − α )

where

n = 47 and α = 0.05