an allergist claims that 50% of ht patients she tests are allergic to some type of weed. what is the probability that exactly 3 of the next 4 patients are allergic to weed

Question

an allergist claims that 50% of the patients she tests are allergic to some type of weed. what is the probability that:

a. exactly 3 of the next 4 patients are allergic to weeds.

b. non of her next 4 patients are allergic to weeds

Answer 1

This is a binomial probability where p=0.5. (p+(1-p))^4=1 represents the sum of all probabilities and is expanded:

p^4+4p^3(1-p)+6p^2(1-p)^2+4p(1-p)^3+(1-p)^4 in which each term relates to a specific occurrence in order:

1. All 4 allergic (1 occurrence)
2. Exactly 3 allergic (4 occurrences)
3. Exactly 2 allergic (6 occurrences)
4. Exactly 1 allergic (4 occurrences)
5. None allergic (1 occurrence)

The coefficients sum to 16 (1+4+6+4+1) because there are 16 possible outcomes.

a. Exactly 3 is 4p^3(1-p)=4(0.5)^3(0.5)=4*(0.5)^4=0.25 (1/4 or 4 out of 16)

b. None is (1-0.5)^4=0.0625 (1/16 or 1 out of 16)

Because the probability of allergy is the same as that for non-allergy, like heads or tails in coins, we can say exactly the same probabilities for the reverse case: exactly 3 non-allergic and all allergic.