Math Problem Statement
Solution
Let's work through each part of this problem based on the probability distribution table provided:
(a) Probability that one or more adults live with a disability:
The probability that none of the five adults live with a disability is given by . The probability that one or more adults live with a disability is the complement of this event:
(b) Probability that two or more adults live with a disability:
The probability that two or more adults live with a disability can be found by subtracting the probabilities for zero and one adult living with a disability from 1:
(c) Probability that three or more adults live with a disability:
Similarly, the probability that three or more adults live with a disability can be found by subtracting the probabilities for zero, one, and two adults from 1:
(d) Expected number of adults with a disability:
The expected value is calculated using the formula for the mean of a probability distribution:
Calculating step by step:
(e) Standard deviation of adults with a disability:
The standard deviation is calculated using the formula:
First, compute :
Now, calculate the standard deviation:
Final Answers:
- (a) Probability one or more: 0.778
- (b) Probability two or more: 0.388
- (c) Probability three or more: 0.114
- (d) Expected number : 1.30
- (e) Standard deviation : 0.98
Would you like further details or have any other questions?
Here are 5 related questions to explore:
- What is the probability that exactly 2 adults have a disability?
- How would the expected value change if the sample size increased to 10 adults?
- What does the complement rule in probability represent, and how is it used here?
- How is standard deviation interpreted in this context?
- How could a change in the overall percentage of adults with a disability (from 26%) affect this distribution?
Tip: Always check that the total sum of probabilities in a probability distribution equals 1.
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Math Problem Analysis
Mathematical Concepts
Probability
Binomial Distribution
Expected Value
Standard Deviation
Formulas
Complement Rule: P(one or more) = 1 - P(none)
P(two or more) = 1 - (P(x = 0) + P(x = 1))
P(three or more) = 1 - (P(x = 0) + P(x = 1) + P(x = 2))
Expected value: μ = Σ[x * P(x)]
Standard deviation: σ = √[Σ(x^2 * P(x)) - μ^2]
Theorems
Law of Total Probability
Properties of Expected Value
Properties of Variance and Standard Deviation
Suitable Grade Level
College / University Level
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