Math Problem Statement
For the US population about 20% get the flu. I had a class where 8 people told me that they got the flu. What is the probability that if I pick 32 students (the size of this class) I get 8 or more that got the flu?
Solution
This problem is asking about the probability of getting 8 or more students who contracted the flu in a class of 32, given that about 20% of the population gets the flu. To solve this, we can use the binomial distribution, since we're dealing with a fixed number of trials (32 students) and two possible outcomes for each trial (got the flu or did not get the flu).
Step 1: Define the problem in terms of binomial distribution
The binomial distribution is appropriate here because:
- Each student either got the flu or didn’t (binary outcomes).
- The probability of a student getting the flu is 0.2 (20%).
- The number of trials is 32 students.
- We're interested in the probability of getting 8 or more students who got the flu.
The binomial probability mass function (PMF) is given by:
where:
- (the number of trials),
- (the probability of a student getting the flu),
- is the number of students who got the flu.
Step 2: Set up the probability calculation
We need to find the probability that 8 or more students got the flu, i.e., .
This is equivalent to summing up the probabilities for to , or we can subtract the probability of from 1:
Step 3: Calculate the probability
We can now calculate the cumulative probability and subtract that from 1.
Let me calculate that for you.The probability of getting 8 or more students who contracted the flu in a class of 32 students is approximately 0.302 or 30.2%.
Would you like more details on this, or do you have any questions?
Here are 5 related questions:
- What is the probability of getting exactly 8 students with the flu?
- How does the probability change if the flu rate is 25% instead of 20%?
- What if the class size increases to 40 students? How does that affect the probability?
- Can the binomial distribution be approximated using the normal distribution in this case?
- What is the expected number of students in a class of 32 to get the flu?
Tip: For large sample sizes and probabilities not too close to 0 or 1, the binomial distribution can be approximated using the normal distribution to simplify calculations.
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Math Problem Analysis
Mathematical Concepts
Probability
Binomial Distribution
Formulas
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
P(X ≥ k) = 1 - P(X ≤ k - 1)
Theorems
Binomial Theorem
Suitable Grade Level
Grades 11-12
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