Given the following probability distribution, identify the missing probability that makes [tex]p(x)[/tex] a valid probability distribution.
| x | 0 | 1 | 2 |
|---|-----|-----|-----|
| p(x) | 1 / 4 | ? | 1 / 4 |
A. 1
B. [tex]\frac{1}{2}[/tex]
C. [tex]\frac{1}{4}[/tex]
D. [tex]\frac{2}{3}[/tex]
Answer (2)
The sum of probabilities in a valid probability distribution equals 1.
Set up the equation: 4 1 + p ( 1 ) + 4 1 = 1 .
Solve for p ( 1 ) : p ( 1 ) = 1 − 4 1 − 4 1 .
The missing probability is: 2 1 .
Explanation
Understand the problem We are given a probability distribution with values for x = 0 and x = 2 , and we need to find the missing probability for x = 1 such that the distribution is valid. A valid probability distribution must sum to 1.
Set up the equation Let p ( 1 ) be the missing probability. We know that the sum of all probabilities in a probability distribution must equal 1. Therefore, we have: p ( 0 ) + p ( 1 ) + p ( 2 ) = 1
Substitute known values We are given p ( 0 ) = 4 1 and p ( 2 ) = 4 1 . Substituting these values into the equation, we get: 4 1 + p ( 1 ) + 4 1 = 1
Solve for p(1) Now, we solve for p ( 1 ) :
p ( 1 ) = 1 − 4 1 − 4 1 p ( 1 ) = 1 − 4 2 p ( 1 ) = 1 − 2 1 p ( 1 ) = 2 1
State the final answer Therefore, the missing probability p ( 1 ) is 2 1 .
Examples
Probability distributions are used in many fields, such as finance, insurance, and physics. For example, in finance, they can be used to model the probability of different stock prices. In insurance, they can be used to model the probability of different claim amounts. Understanding how to complete a probability distribution is crucial for making accurate predictions and decisions in these fields.
The missing probability that makes the distribution valid is 2 1 . The calculations show that the sum of the probabilities must equal 1, and by solving for the unknown probability, we find it to be 2 1 . Therefore, the answer is option B, 2 1 .
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