Module 1 - Text

The Calculation of Probability ($Pr(A)$)

The sources define probability as the quantitative measure of uncertainty in situations involving randomness, which is the lack of pattern or predictability in events. Probabilities are always expressed as values ranging from 0 (no chance) to 1 (a certainty).

I. The Fundamental Probability Formula

The basic method for calculating the probability of an event $A$, denoted $Pr(A)$, is given by the ratio:

$$Pr(A)=\frac{n(A)}{N}$$

This calculation requires two essential counts:

1. $n(A)$: The number of ways in which the event $A$ can occur.
2. $N$: The total number of possible outcomes.

Crucially, this formula is valid only when each outcome is equally likely.

Example (Single Die): If tossing a fair die, $N=6$ possible outcomes $1,2,3,4,5,6$. For the event $A$ (tossing a 6), $n(A)=1$. Therefore, $Pr(6)=1/6$.

II. Determining Total Outcomes ($N$): The Counting Principle

When an experiment or situation consists of multiple steps or stages, the Basic Counting Principle (or multiplication principle) is used to find $N$, the total number of possible outcomes.

If a situation has $k$ stages, with $n_1$ choices at the first step, $n_2$ choices at the second step, and so on up to $n_k$ choices at the last step, the total number of possible outcomes ($N$) is the product:

$$N=n_1\times n_2\times \cdots \times n_k$$

Example (Two Dice): Tossing two dice involves two stages, each with $n_1=6$ and $n_2=6$ choices. The total number of possible outcomes is $N=6\times 6=36$.

This counting method is critical because while the individual outcomes of the dice are equally likely, the sum of the two dice is not. For the event $A$ (tossing a sum of 4), $n(A)=3$ ways (1,3; 2,2; and 3,1). Therefore, $Pr(4)=3/36=1/12$.

III. Calculating Probability in Statistical Distributions

The basic probability calculation extends directly into the concept of a Probability Distribution, which pairs the possible values of a Random Variable ($X$) with their corresponding probability $Pr(X=x)$. The sum of all probabilities in any distribution must equal 1.

1. Discrete Distributions (Summing Probabilities): For discrete variables (where outcomes can be listed, like die rolls or the number of girls in a family), the probability of a range of outcomes is found by summing the individual probabilities. For example, the probability of rolling an even number is $Pr(\text{even})=Pr(2)+Pr(4)+Pr(6)=1/6+1/6+1/6=1/2$.
2. Continuous Distributions (Area Under Curve): For continuous variables (like tail lengths or ages), the shaded area under the distribution curve represents a proportion of all values in that distribution, which is equivalent to the probability of an event of interest. This allows finding the probability that a randomly selected value lies within a specific range.
3. Binomial Calculation (Formula): For specific discrete models like the Binomial distribution, the probability of $x$ successes in $n$ trials is calculated using the formula:

$$Pr(X=x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$$

The goal of calculating $Pr(A)$ is often to determine the expected value or proportion of an event in the long term. For example, the expected number of times an event will occur is calculated by multiplying its probability by the total number of trials or items.