Which of the following is a valid probability distribution?

• A. $P\left(1\right)=0.4$, $P\left(2\right)=0.4$, $P\left(3\right)=0.4$
• B. $P\left(1\right)=0.3$, $P\left(2\right)=0.3$, $P\left(3\right)=0.1$
• C. $P\left(1\right)=0.2$, $P\left(2\right)=0.5$, $P\left(3\right)=0.3$
• D. $P\left(1\right)=-0.1$, $P\left(2\right)=0.6$, $P\left(3\right)=0.5$

Answer: (C)

Verified step by step guidance

1. Recall the two main conditions for a valid probability distribution: (1) Each probability must be between 0 and 1 inclusive, i.e., \$0 \leq P(x) \leq 1\( for all \)x\(, and (2) the sum of all probabilities must equal 1, i.e., \)\sum P(x) = 1$.
2. Check the first distribution: \(P(1) = 0.4\), \(P(2) = 0.4\), \(P(3) = 0.4\). Verify if each probability is between 0 and 1, then sum them up to see if they equal 1.
3. Check the second distribution: \(P(1) = 0.3\), \(P(2) = 0.3\), \(P(3) = 0.1\). Confirm each probability is valid and sum them to check if the total is 1.
4. Check the third distribution: \(P(1) = 0.2\), \(P(2) = 0.5\), \(P(3) = 0.3\). Again, verify the range of each probability and sum them to confirm if they add up to 1.
5. Check the fourth distribution: \(P(1) = -0.1\), \(P(2) = 0.6\), \(P(3) = 0.5\). Notice that probabilities cannot be negative, so this distribution violates the first condition and is invalid.