Probability

Engineering Context

Since engineers often deal with the analysis of physical events, most disciplines are at least somewhat related to testing. Tests and experiments in engineering contain uncertainties, which will affect the results of an experiment, even with extreme precision on the part of the engineer conducting the studies. Probability theory was developed to understand and quantify this uncertainty so that engineers can understand the limits of what they know and develop.

MAE: The probability that a pressure test results in a value of 4 MPa

P 4 = 0.2

ECE: The probability of getting a defective computer chip

P defective = 0.02

BENG: The probability that a tissue sample provides useful data

P useful = 0.5

CEE: The probability that a bridge buckles in the presence of flooding

P failure = 0.001

The Essentials

The probability that a specific outcome in an event occurs can be noted as

P O i = p i

where Oi is the outcome and pi is its probability. Probabilities are defined to satisfy the constraints

0 ≤ p i ≤ 1

and

∑ i=1 n p i = 1

A Deeper Dive

When dealing with experiments and uncertainties, we can group the possible outcomes into a sample space, which we will label S. As an example, when rolling two six-sided die, the sample or state space is described by

S = i j  for all  i,  j   ∈   I

where i is the number on the first die and j is the number on the second die. This sample space can be visualized as

1 1 1 2 … 1 6 2 1 2 2 … 2 6 ⋮ ⋮ ⋱ ⋮ 6 1 6 2 … 6 6

Assuming that we have two fair dice that are equally likely to land on any given side, the probability that we roll first a 1 and then a 6 is then denoted

P 1 6 = 1 / 36 = 0.0277

since there is exactly one instance where we have this outcome out of the total number of potential combinations (6×6 = 36) of die rolls. In contrast, the probability that we roll a 6 in either of the rolls is

P 6 = 11 / 36 = 0.305

which corresponds to the last row and last column of the sample space shown above (6×2−1 = 11 to not count (6, 6) twice). From this, we can see that we have a higher probability of rolling a 6 in either of our rolls than getting a specific combination of rolls, as explored in the first example, which is pretty intuitive!

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