The blood types B- and AB- are the rarest of the eight huan blood types representing 1.5% and .6% respectively.
a) If the blood types of a random sample of 1000 blood donors are recorded, what is the probability that 10 or more will be AB-?
b) If the blood types of a random sample of 1000 blood donors are recorded, what is the probability that 20 t 40 inclusive of the samples are B-?
c) If a particular city had a blood drive and 3% of the donations were B-, would we have reason to believe that this town has a higher than normal number of donors who are B-? (Hint: Calculate the probablilty of 15 or more donors being B- for a random sample of 500 and then discuss the probability obtained.)
I suggest that you use the methodology suggested by Mathguru in your next post.
To solve these probability questions, we can use the binomial probability formula. The formula for the probability of obtaining exactly k successes in n independent Bernoulli trials is:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes
- n C k is the number of possible combinations of k successes out of n trials
- p is the probability of success in each trial
- (1-p) is the probability of failure in each trial
- k is the number of successes
- n is the total number of trials
We can use this formula to solve each question:
a) Probability that 10 or more will be AB-:
In this case, k = 10, n = 1000, and p = 0.006 (probability of AB-). We need to calculate the sum of the probabilities for k = 10, 11, 12, ..., 1000:
P(X ≥ 10) = P(X = 10) + P(X = 11) + .... + P(X = 1000)
Using the binomial probability formula mentioned above, we can calculate each individual probability and sum them up. This will give us the probability that 10 or more blood donors will be AB-.
b) Probability that 20 to 40 inclusive of the samples are B-:
In this case, k ranges from 20 to 40, n = 1000, and p = 0.015 (probability of B-). We need to calculate the sum of the probabilities for k = 20, 21, 22, ..., 40:
P(20 ≤ X ≤ 40) = P(X = 20) + P(X = 21) + .... + P(X = 40)
Using the same binomial probability formula, we can calculate each individual probability and sum them up. This will give us the probability that the number of B- blood donors falls within the range of 20 to 40 (inclusive).
c) Probability of 15 or more donors being B- for a random sample of 500:
In this case, k = 15, n = 500, and p = 0.03 (probability of B-). We similarly calculate the probability using the binomial formula:
P(X ≥ 15) = P(X = 15) + P(X = 16) + .... + P(X = 500)
By calculating this probability, we can determine if the city's blood drive had a higher than normal number of donors who are B-.
It's important to note that these calculations require a good understanding of binomial probability and possibly the use of a calculator or statistical software to calculate the combination values.