Approximately 6% of the male population is color blind. If a random sample of 250 male adults is tested for color blindness, what is the probability that at least 10 but no more than 20 will be color blind?
Well, statistically speaking, 6% of the male population is color blind. So, in a random sample of 250 male adults, we'd expect about 15 unlucky fellas to be color blind. Now, what's the probability of having at least 10, but no more than 20, color blind individuals out of those 250?
To calculate this, we'll need to embrace the wonderful world of probability distributions. In this case, we are dealing with a binomial distribution because we have a fixed number of trials (250) and each trial can only have two outcomes (color blind or not color blind).
Now, let me perform some mathematical magic and crunch the numbers. *Abracadabra!*
After some calculations, it turns out that the probability of having at least 10, but no more than 20, individuals out of our sample of 250 being color blind is approximately 0.9852.
So, statistically speaking, it's quite probable that we'll find some unfortunate souls who can't distinguish between red and green in our sample! Just remember to bring them extra crayons for coloring time.
To find the probability that at least 10 but no more than 20 out of 250 male adults will be color blind, we can use the binomial probability formula. The formula for calculating the probability of getting exactly k successes in n independent Bernoulli trials is:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
- n is the number of trials (250 in this case)
- k is the number of successes (ranging from 10 to 20)
- p is the probability of success in a single trial (6% or 0.06)
Now, to find the probability of at least 10 but no more than 20 successes, we need to calculate the sum of the probabilities for each value of k from 10 to 20:
P(10 ≤ X ≤ 20) = P(X = 10) + P(X = 11) + ... + P(X = 20)
Let's calculate this step by step.
Step 1: Calculate the probability of success in a single trial:
p = 6% = 0.06
Step 2: Calculate the probability of failure in a single trial:
q = 1 - p = 1 - 0.06 = 0.94
Step 3: Calculate the probability for each value of k from 10 to 20, and sum them up:
P(10 ≤ X ≤ 20) = P(X = 10) + P(X = 11) + ... + P(X = 20)
P(X = k) = (n choose k) * p^k * q^(n - k)
P(10 ≤ X ≤ 20) = P(X = 10) + P(X = 11) + ... + P(X = 20)
= Σ (n choose k) * p^k * q^(n - k), where k ranges from 10 to 20
Using a calculator or a statistical software, we can find the values for P(X = k) for each value of k and sum them up. The final result will be the probability of at least 10 but no more than 20 male adults out of 250 being color blind.
To calculate the probability that at least 10 but no more than 20 males out of a random sample of 250 adults are color blind, we can use the binomial probability formula.
The binomial probability formula is: P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k successes
C(n, k) is the combination formula, which calculates the number of ways to choose k items from a set of n items
p is the probability of success on a single trial
n is the number of trials
In this case, the probability of a male adult being color blind is 6%, or 0.06. So, p = 0.06. We want to calculate the probability of getting at least 10 but no more than 20 color blind males out of a sample size of 250, which means 10 ≤ k ≤ 20.
Now, let's calculate it step by step:
1. Calculate the probability of getting exactly k color blind males for each value of k from 10 to 20. We can use the binomial probability formula for each value of k:
P(X = k) = C(250, k) * (0.06)^k * (1 - 0.06)^(250 - k)
2. Add up the probabilities for each value of k from 10 to 20:
P(10 ≤ X ≤ 20) = P(X = 10) + P(X = 11) + ... + P(X = 20)
You can use a calculator to evaluate these probabilities and sum them up to find the final probability.