Which of the following represents a valid probability distribution?

• A. A distribution where $P\left(A\right)=0.3$, $P\left(B\right)=0.5$, and $P\left(C\right)=0.2$
• B. A distribution where $P\left(A\right)=-0.1$, $P\left(B\right)=1.1$
• C. A distribution where $P\left(A\right)=0.6$, $P\left(B\right)=0.6$
• D. A distribution where $P\left(A\right)=0.4$, $P\left(B\right)=0.3$, and $P\left(C\right)=0.1$

Answer: (A)

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(A) \leq 1\( for every event \)A\(, and (2) the sum of all probabilities must equal 1, i.e., \)\sum P(A_i) = 1\( where \)A_i$ are all possible events.
2. Check each given distribution to ensure all probabilities are within the range 0 to 1. If any probability is less than 0 or greater than 1, that distribution is invalid.
3. For each distribution, add up all the probabilities. Use the formula \(\sum P(A_i) = P(A) + P(B) + P(C) + \dots\) depending on the number of events given.
4. Verify if the sum of probabilities equals exactly 1. If the sum is less than or greater than 1, the distribution is not valid.
5. Identify the distribution that meets both criteria: all probabilities between 0 and 1, and the sum of probabilities equal to 1. This distribution represents a valid probability distribution.