Math Problem Statement
Detailed steps
Solution
Let's solve this problem step-by-step.
Part (a)
We are given:
- The probability of a patient dying after the surgery, .
- The probability of a patient surviving, .
- We want to find the probability that all 8 patients survive, so we’re looking for the probability of 0 fatalities out of 8 patients.
This scenario can be modeled using the Binomial Distribution where:
- (number of patients),
- (no fatalities),
- .
The probability of 0 fatalities (all 8 patients survive) is calculated as:
Calculating:
So, the probability of treating 8 patients without a single fatality is approximately 0.1001 or 10.01%.
Part (b)
Here, we have:
- A 1% chance of dying from a certain illness in a year, so .
- The group size is .
-
Probability that fewer than 5 people die:
Since is large and is small, we can use the Poisson approximation to the Binomial distribution. In this approximation, the parameter (mean number of occurrences) is calculated as:
We are asked for the probability that fewer than 5 people die, which is the cumulative probability for in a Poisson distribution with .
Calculating this: where .
Let's calculate each term:
Summing these probabilities:
Therefore, the probability that fewer than 5 people die is approximately 0.6289 or 62.89%.
-
Variance of the number dying:
For a Binomial distribution, the variance is given by:
So, the variance of the number of people dying in a year is approximately 3.96.
Would you like further explanations on any part, or do you have questions about the method used?
Related Questions for Further Exploration
- How would the probability change in part (a) if the mortality rate increased to 30%?
- What if we wanted the probability of treating 8 patients with at least one fatality in part (a)?
- How would the Poisson approximation change if we considered only a group of 100 people in part (b)?
- In part (b), how would the probability change if the mortality risk was 2% instead of 1%?
- What is the expected number of deaths in the group of 400 people for part (b)?
Tip
When dealing with large and small ( p
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Math Problem Analysis
Mathematical Concepts
Binomial Distribution
Poisson Approximation
Probability Theory
Formulas
Binomial Probability Formula: P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Poisson Approximation: P(X = k) = (λ^k * e^(-λ)) / k!
Variance of a Binomial Distribution: Variance = n * p * (1 - p)
Theorems
Binomial Distribution Theorem
Poisson Approximation to the Binomial Distribution
Suitable Grade Level
Undergraduate Level
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