Question:
Given the random variable \( X \) which takes the values 0, 1, 2, 7, 11, and 12 with the following probabilities:
\[
P(X = 0) = 0.4, \quad P(X = 1) = 0.3, \quad P(X = 2) = 0.1, \quad P(X = 7) = 0.1, \quad P(X = 11) = ?
\]
Given the random variable \( X \) which takes the values 0, 1, 2, 7, 11, and 12 with the following probabilities:
\[
P(X = 0) = 0.4, \quad P(X = 1) = 0.3, \quad P(X = 2) = 0.1, \quad P(X = 7) = 0.1, \quad P(X = 11) = ?
\]
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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.
Updated On: Feb 17, 2025
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Solution and Explanation
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 \( 0 \).
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