Question

Given the random variable \( X \) which takes the values 0, 1, 2, 7, 11, and 12 with the following probabilities: \[ P(X = 0) = 0.4, \quad P(X = 1) = 0.3, \quad P(X = 2) = 0.1, \quad P(X = 7) = 0.1, \quad P(X = 11) = ? \]

Hint:

In any probability distribution, remember that the sum of all probabilities must equal 1. If any probabilities are missing, use this rule to solve for the unknown probabilities.

Answer 1

Since the total probability for any probability distribution must sum to 1, we can calculate \( P(X = 12) \) by using the equation: \[ P(X = 0) + P(X = 1) + P(X = 2) + P(X = 7) + P(X = 11) + P(X = 12) = 1 \] Substituting the known values: \[ 0.4 + 0.3 + 0.1 + 0.1 + 0.1 + P(X = 12) = 1 \] Simplifying the left-hand side: \[ 1.0 + P(X = 12) = 1 \] Solving for \( P(X = 12) \): \[ P(X = 12) = 1 - 1.0 = 0 \] Thus, the probability \( P(X = 12) \) is \( 0 \).