In testing a new drug, researchers found that 10% of all patients using it will have a mild side effect. A random sample of 14 patients using the drug is selected. Find the probability that:
(A) exactly two will have this mild side effect
(B) at least three will have this mild side effect.
Hmmmmmm.....not sure.....
To find the probability, we can use the binomial probability formula. The formula for the probability of exactly r successes in n trials, where the probability of success in each trial is p, is given by:
P(r) = C(n,r) * p^r * (1-p)^(n-r)
where C(n,r) represents the number of combinations of n items taken r at a time, and p represents the probability of success in each trial.
In this case, the probability of having a mild side effect is 10%, which can be expressed as 0.10. So, the probability of not having a mild side effect is 1 - 0.10 = 0.90.
(A) To find the probability of exactly two patients having this mild side effect, we can use the formula mentioned above:
P(2) = C(14,2) * 0.10^2 * 0.90^(14-2)
where C(14,2) = 14! / (2! * (14-2)!), which simplifies to 91.
Plugging in the values:
P(2) = 91 * 0.10^2 * 0.90^12 ≈ 0.302
So, the probability is approximately 0.302, or 30.2%.
(B) To find the probability of at least three patients having this mild side effect, we need to find the probability of three, four, five, ..., fourteen patients having the side effect and add them up.
P(at least 3) = P(3) + P(4) + P(5) + ... + P(14)
We can use the formula mentioned above to calculate each term and then sum them up.
P(at least 3) ≈ P(3) + P(4) + P(5) + ... + P(14)
Calculating each term:
P(3) = C(14,3) * 0.10^3 * 0.90^(14-3)
P(4) = C(14,4) * 0.10^4 * 0.90^(14-4)
...
P(14) = C(14,14) * 0.10^14 * 0.90^(14-14)
Summing up the terms will give us the desired probability.
Note: You can use a calculator or a software program to calculate the combinations.