Of the volunteers donating blood in a clinic, the true population proportion that havethe Rhesus factor present in their blood is 0.8.(a) [2 marks] If 20 volunteers are randomly selected, find the probability that exactly15 volunteers have the Rhesus factor
Question
Of the volunteers donating blood in a clinic, the true population proportion that havethe Rhesus factor present in their blood is 0.8.(a) [2 marks] If 20 volunteers are randomly selected, find the probability that exactly15 volunteers have the Rhesus factor
Solution
This problem can be solved using the binomial probability formula, which is:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
where:
- P(X=k) is the probability of k successes in n trials
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success on a single trial
- n is the number of trials
- k is the number of successes
In this case:
- n = 20 (the number of volunteers)
- k = 15 (the number of volunteers with the Rhesus factor)
- p = 0.8 (the probability of a volunteer having the Rhesus factor)
Substituting these values into the formula gives:
P(X=15) = C(20, 15) * (0.8^15) * ((1-0.8)^(20-15))
First, calculate C(20, 15), which is the number of combinations of 20 items taken 15 at a time. This can be calculated as:
C(20, 15) = 20! / [(20-15)! * 15!] = 15504
Next, calculate (0.8^15) = 0.035184
Then, calculate ((1-0.8)^(20-15)) = 0.00032
Finally, substitute these values back into the formula to get:
P(X=15) = 15504 * 0.035184 * 0.00032 = 0.174
So, the probability that exactly 15 out of 20 randomly selected volunteers have the Rhesus factor is approximately 0.174, or 17.4%.
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