In a region, 90% of the population have brown eyes. If 15 people are randomly selected, find the probability that at
least 13 of them have brown eyes. Is it
unusual to randomly select 15 people and find that at least 13 of them have brown eyes? Why or why not?
To find the probability that at least 13 of the 15 randomly selected people have brown eyes, we can use the binomial probability formula.
The formula for the probability of X successes in n trials, where the probability of success is p, is:
P(X = x) = (nCx)(p^x)((1-p)^(n-x))
Given that 90% of the population has brown eyes, which means the probability of selecting a person with brown eyes is 0.9, and that we want to find the probability of at least 13 out of 15 people having brown eyes, we need to calculate the probability of exactly 13 brown-eyed people, 14 brown-eyed people, and 15 brown-eyed people.
P(X ≥ 13) = P(X = 13) + P(X = 14) + P(X = 15)
P(X = 13) = (15C13)(0.9^13)(0.1^2)
P(X = 14) = (15C14)(0.9^14)(0.1^1)
P(X = 15) = (15C15)(0.9^15)(0.1^0)
To calculate the combination, (nCx), we use the formula:
nCx = n! / (x!(n-x)!)
Now, let's calculate each probability:
P(X = 13) = (15C13)(0.9^13)(0.1^2)
= (15! / (13!(15-13)!)) * (0.9^13) * (0.1^2)
= 105 * 0.2541865828329002 * 0.01
= 0.26772062300000006
P(X = 14) = (15C14)(0.9^14)(0.1^1)
= (15! / (14!(15-14)!)) * (0.9^14) * (0.1^1)
= 15 * 0.22876792 * 0.1
= 0.34315188000000005
P(X = 15) = (15C15)(0.9^15)(0.1^0)
= (15! / (15!(15-15)!)) * (0.9^15) * (0.1^0)
= 1 * 0.2058911321 * 1
= 0.2058911321
Now, let's calculate the total probability:
P(X ≥ 13) = P(X = 13) + P(X = 14) + P(X = 15)
= 0.26772062300000006 + 0.34315188000000005 + 0.2058911321
= 0.8167636351000001
The probability of randomly selecting 15 people and finding that at least 13 of them have brown eyes is approximately 0.8168, or 81.68%.
To determine if this is unusual or not, we need to consider the significance level. If the significance level is set at 5%, then any event with a probability less than 5% would be considered unusual. In this case, the probability of at least 13 out of 15 people having brown eyes is higher than 5%, so it would not be considered unusual.
Therefore, it is not unusual to randomly select 15 people and find that at least 13 of them have brown eyes.
To find the probability that at least 13 people out of the 15 randomly selected have brown eyes, we can use the binomial probability formula:
P(X ≥ k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X ≥ k) is the probability of getting k or more successes (in this case, people with brown eyes),
- n is the number of trials (number of people selected),
- k is the number of successful outcomes (number of people with brown eyes),
- p is the probability of success in a single trial (probability of a person having brown eyes).
In this scenario, n = 15, k = 13 or 14 or 15 (to find the probability of at least 13 people having brown eyes), and p = 0.9 (probability of a person having brown eyes).
Now, let's calculate the probability for each case and sum them up:
P(X ≥ 13) = P(X = 13) + P(X = 14) + P(X = 15)
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Using this formula, we can calculate the individual probabilities and sum them up to find the final probability.
As for whether it is unusual to randomly select 15 people and find that at least 13 of them have brown eyes, it depends on what is considered unusual in this context. With a population where 90% have brown eyes and only 15 people being randomly selected, it would actually be quite likely to find at least 13 people with brown eyes. However, the specific threshold for what is considered unusual may vary depending on the context and the specific criteria being evaluated.
P(X ≥ 13) = P (X= 13) + P (X = 14) + P (X = 15)
P(X ≥ 13) =(15C13)(.9)^13(.1)^2+ (15C14)(.9)^14(.1)^1+ (15C15)(.9)^15(.1)^0
= .8159