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 even sure....
To solve this problem, we can use the concept of probability and the binomial distribution. The binomial distribution is applicable when we have a fixed number of independent trials (in this case, the 14 patients) and each trial has only two possible outcomes (in this case, having or not having the mild side effect).
Before we calculate the probabilities, let's define a few terms:
- n: the total number of trials or patients (14 in this case)
- p: the probability of success (having the mild side effect) in a single trial (10% or 0.10 in this case)
- x: the number of successful trials we are interested in (2 in part A and at least 3 in part B)
Now, let's calculate the probabilities:
(A) Exactly two will have this mild side effect:
The probability of exactly x successes in n trials can be calculated using the binomial probability formula:
P(X = x) = (nCx) * p^x * (1-p)^(n-x)
In this case, we have n = 14, x = 2, and p = 0.10. Plugging these values into the formula, we get:
P(X = 2) = (14C2) * (0.10)^2 * (1-0.10)^(14-2)
Calculating (14C2) using the combination formula, we find:
(14C2) = (14! / (2! * (14-2)!)) = 91
Now, we can substitute these values into the formula and calculate the probability.
P(X = 2) = 91 * (0.10)^2 * (0.90)^12
(B) At least three will have this mild side effect:
To find the probability of at least three successes, we need to calculate the probability of having exactly three, exactly four, and so on, until 14. Then, we add up all those probabilities.
P(X ≥ 3) = P(X = 3) + P(X = 4) + ... + P(X = 14)
We can repeat the above steps for each value of x from 3 to 14 and add them together to get the probability.
Remember that for each calculation, you'll need to use the binomial probability formula as described above.
I hope this helps clarify how to approach this problem. If you need any further assistance, feel free to ask!