In Problems 15 and 16, determine the required value of the missing probability to make the distribution a discrete probability distribution.

Verified step by step guidance

1. Step 1: Understand that for a discrete probability distribution, the sum of all probabilities must equal 1. This is a fundamental property of probability distributions.
2. Step 2: Identify the known probabilities from the table: \(P(3) = 0.4\), \(P(5) = 0.1\), and \(P(6) = 0.2\). The missing probability is \(P(4)\), which we need to find.
3. Step 3: Set up the equation representing the sum of all probabilities: \(P(3) + P(4) + P(5) + P(6) = 1\).
4. Step 4: Substitute the known values into the equation: \$0.4 + P(4) + 0.1 + 0.2 = 1$.
5. Step 5: Solve for \(P(4)\) by combining the known probabilities and subtracting from 1: \(P(4) = 1 - (0.4 + 0.1 + 0.2)\).

Key Concepts

Discrete Probability Distribution

A discrete probability distribution assigns probabilities to each possible value of a discrete random variable. Each probability must be between 0 and 1, and the sum of all probabilities must equal 1 to represent a valid distribution.

Sum of Probabilities Equals One

In any probability distribution, the total of all individual probabilities must be exactly 1. This ensures that the distribution accounts for all possible outcomes of the random variable.

Finding Missing Probability

To find a missing probability in a distribution, subtract the sum of the known probabilities from 1. The result is the value needed to complete the distribution so that the total probability sums to 1.