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....
No problem! Let's break it down step by step.
To solve these probabilities, we will use the binomial probability formula. The formula for finding the probability of exactly k successes in n trials is:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials
- k is the number of successes
- p is the probability of success in a single trial
- (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials
For part (A), we need to find the probability that exactly two patients from the sample of 14 will have the mild side effect. Given that 10% of all patients using the drug will have this side effect, the probability of success in a single trial (p) is 0.10.
Let's substitute these values into the formula:
P(X = 2) = (14 choose 2) * 0.10^2 * (1-0.10)^(14-2)
Now we can calculate it step by step:
P(X = 2) = (14! / (2! * (14-2)!)) * 0.10^2 * (0.90)^12
P(X = 2) = (14 * 13 / (2 * 1)) * 0.01 * 0.282429536
P(X = 2) = 91 * 0.00282429536
P(X = 2) ≈ 0.2572
So, the probability that exactly two patients will have the mild side effect is approximately 0.2572.
Now let's move on to part (B), which asks for the probability of at least three patients having the mild side effect. To find this probability, we need to calculate the sum of probabilities for three or more patients having the side effect.
We can start by finding the individual probabilities:
P(X ≥ 3) = P(X = 3) + P(X = 4) + ... + P(X = 14)
However, this might be a bit tedious to calculate manually, so we can simplify it by finding the complement of having fewer than three cases:
P(X ≥ 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]
Let's calculate it step by step:
P(X ≥ 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]
P(X ≥ 3) = 1 - [(14 choose 0) * 0.10^0 * (0.90)^14 + (14 choose 1) * 0.10^1 * (0.90)^13 + (14 choose 2) * 0.10^2 * (0.90)^12]
Now we can substitute and calculate:
P(X ≥ 3) = 1 - [1 * 1 * 0.282429536 + 14 * 0.10 * 0.282429536 + 91 * 0.01 * 0.282429536]
P(X ≥ 3) = 1 - [0.282429536 + 0.282429536 + 0.257229523]
P(X ≥ 3) = 1 - 0.822088595
P(X ≥ 3) ≈ 0.1779
Therefore, the probability that at least three patients will have the mild side effect is approximately 0.1779.
I hope this explanation helps! Let me know if you have any further questions.
No worries! Let's break down the problem and start by understanding some basic concepts.
In this scenario, we have a random sample of patients using a new drug, and we want to calculate the probability of certain events happening. To solve these types of probability problems, we typically use binomial probability.
Binomial probability is used when we have a fixed number of independent trials/outcomes (in this case, patients using the drug), and each trial can have one of two possible outcomes (in this case, either experiencing the mild side effect or not). We also assume that the probability of success (experiencing the side effect) remains constant for each trial.
Now, let's proceed to solve the problem.
(A) To find the probability that exactly two patients will have the mild side effect, we can use the binomial probability formula:
P(X = k) = (nCk) * p^k * (1 - p)^(n-k),
where n is the number of trials (sample size), k is the number of successes (patients experiencing the side effect), p is the probability of success (probability of having the side effect in this case), and nCk represents the number of combinations of n items taken k at a time.
In this case, n = 14 (sample size) and p = 0.10 (probability of each patient having the mild side effect).
Plugging these values into the formula, we get:
P(X = 2) = (14C2) * 0.10^2 * (1 - 0.10)^(14-2).
Calculating the expression, we find the probability that exactly two patients will have the mild side effect.
(B) To find the probability that at least three patients will have the mild side effect, we can calculate the complementary probability. That is, we find the probability that fewer than three patients will have the side effect and subtract it from 1.
So, P(X ≥ 3) = 1 - P(X < 3).
Using the binomial probability formula, we can calculate P(X < 3) by adding the probabilities of having zero and one patient with the side effect:
P(X<3) = P(X=0) + P(X=1) = (14C0) * 0.10^0 * (1 - 0.10)^(14-0) + (14C1) * 0.10^1 * (1 - 0.10)^(14-1).
Once we have that value, we can subtract it from 1 to find the probability that at least three patients will have the mild side effect.
I hope this explanation helps you understand how to calculate these probabilities!