Blood type AB is found in only 3% of the population†. If 240 people are chosen at random, find the probability of the following. (Round your answers to four decimal places.)
(a) 5 or more will have this blood type
(b)between 5 an 10 will have this blood type
To find the probabilities, we can use the binomial probability formula:
P(x) = (n C x) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials (240 people)
x is the number of successes (in this case, the number of people with blood type AB)
p is the probability of success in a single trial (3% or 0.03)
(1-p) is the probability of failure in a single trial (97% or 0.97)
Let's calculate each probability step-by-step:
(a) To find the probability that 5 or more people will have blood type AB:
P(5 or more) = P(5) + P(6) + P(7) + ... + P(240)
P(x) = (240 C x) * 0.03^x * 0.97^(240-x)
P(5 or more) = P(5) + P(6) + P(7) + ... + P(240)
P(5 or more) = Σ[P(x)] for x = 5 to 240
To calculate this sum, it would be time-consuming to manually calculate each term. So, we can use a calculator or software to find the answer directly. Let's assume we have a calculator that can handle these calculations.
(b) To find the probability that between 5 and 10 people (inclusive) will have blood type AB:
P(5 to 10) = P(5) + P(6) + P(7) + P(8) + P(9) + P(10)
P(5 to 10) = Σ[P(x)] for x = 5 to 10
Again, we can use a calculator or software to find the answer directly.
Note: I cannot provide the final numerical values without using a calculator or software.
To find the probability of these events, we can use the binomial probability formula. The formula for the probability of obtaining exactly k successes in n independent trials is:
P(k) = C(n, k) * p^k * (1-p)^(n-k)
where:
P(k) is the probability of exactly k successes
n is the number of trials (in this case, the number of people chosen at random)
k is the number of successes (in this case, the number of people with blood type AB)
p is the probability of success (in this case, the probability of choosing a person with blood type AB)
C(n, k) is the number of combinations of n items taken k at a time, which can be calculated using the formula: C(n, k) = n! / (k!(n-k)!)
Let's calculate the probabilities:
(a) 5 or more will have this blood type:
To find the probability of 5 or more people having blood type AB, we need to calculate the probabilities for 5, 6, 7, 8, ..., 240 people having this blood type, and then add them together.
P(5 or more) = P(5) + P(6) + P(7) + ... + P(240)
We can use a calculator or a computer program to perform this calculation. One way to do it is using a spreadsheet.
(b) Between 5 and 10 will have this blood type:
To find the probability of between 5 and 10 people having blood type AB, we need to calculate the probabilities for 5, 6, 7, 8, 9, and 10 people having this blood type, and then add them together.
P(between 5 and 10) = P(5) + P(6) + P(7) + P(8) + P(9) + P(10)
Again, we can use a calculator or a computer program to perform this calculation.
Here's one way to do this problem:
n = 240
p = .03
q = 1 - p = 1 - .03 = .97
For (a), you will need to find P(0) through P(4). Add together, then subtract the total from 1 for the probability.
I'll let you try (b).
You can use a binomial probability table or calculate by hand using the following formula:
P(x) = (nCx)(p^x)[q^(n-x)]
I hope this will help get you started.