12.4. Example of PMF

Example 1: Tossing a fair coin once Let 𝑋 be a random variable representing the outcome: • 𝑋 = 0 → Tails • 𝑋 = 1 → Heads 𝑃 (𝑥) = {0.5 if 𝑥 = 0 0.5 if 𝑥 = 1 0 otherwise

In this example, we define a random variable X that can take on two values based on a coin toss. When we say X = 0 represents tails and X = 1 represents heads, we are assigning specific outcomes to numerical values. The Probability Mass Function (PMF) specifies that the probability of getting tails (P(X=0)) is 0.5, and the probability of getting heads (P(X=1)) is also 0.5. The PMF values are zero for any outcomes not equal to 0 or 1, which reflects that these are the only possible outcomes for a fair coin toss.

Imagine you're flipping a coin with a friend. Each time you flip the coin, it can either land on heads or tails. If you're keeping track of the results, you notice that you get heads about half the time and tails about half the time, matching the PMF's predictions. This is a straightforward, everyday example of using probability.

Example 2: Rolling a fair 6-sided die 𝑃 𝑋(𝑥) = {1/6 for 𝑥 = 1,2,3,4,5,6 0 otherwise

This example introduces another discrete random variable X represented by the outcome of rolling a fair 6-sided die. Here, the random variable can take on six different values: 1, 2, 3, 4, 5, or 6. The PMF indicates that the probability of rolling any one of these numbers is 1/6, due to the symmetry and fairness of the die. For any other number that is not between 1 and 6, the probability is zero. This uniformly distributions reflects the equal likelihood of each outcome when rolling the die.

When you roll a die during a game night, you know there's an equal chance for each of the six faces to land face up. This is a common scenario in board games, and it illustrates the concept of PMF well since each outcome's probability is the same. Just like when the die is fair, the PMF tells you that rolling a three, for instance, is equally likely as rolling any other number.

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