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.....
I'm sorry, Jennifer -- but I've forgotten everything I learned in the statistics class that I had 40 years ago.
oh...its okay...do u know who can help me? no one seems to be helping me :(
PsyDag gets a lot of statistics questions, but he usually answers questions a couple of hour a day. The same is true of some of our other math tutors.
okay thank u
To solve these probability questions, we can use the binomial probability formula. The formula is as follows:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
where:
- P(X = k) is the probability of getting exactly k successes
- n is the number of trials or sample size
- k is the number of desired successes
- (n C k) represents the combination of n items taken k at a time (also known as binomial coefficient)
- p is the probability of a success in one trial
- (1-p) is the probability of a failure in one trial
- ^ represents exponentiation
Now, let's solve the given probability questions step-by-step:
(A) Probability of exactly two patients having the mild side effect:
In this case:
- n (sample size) = 14 (as given in the question)
- k (desired successes) = 2 (as specified in the question)
- p (success rate) = 0.10 (as given in the question)
Let's substitute these values into the binomial probability formula:
P(X = 2) = (14 C 2) * 0.10^2 * (1-0.10)^(14-2)
To simplify further, we need to calculate the combination (14 C 2), which can be calculated as follows:
(14 C 2) = (14!) / [(2!)(14-2)!]
= (14*13) / (2*1)
= 91
Substituting this value into the formula:
P(X = 2) = 91 * 0.10^2 * (1-0.10)^(14-2)
= 91 * 0.01 * (0.9)^12
P(X = 2) ≈ 0.2413
Therefore, the probability of exactly two patients having the mild side effect is approximately 0.2413.
(B) Probability of at least three patients having the mild side effect:
In this case, we need to find the cumulative probability of having three, four, five, ..., up to fourteen patients with the mild side effect. We'll then sum up these probabilities to get the desired result.
P(at least three) = P(X ≥ 3) = P(X = 3) + P(X = 4) + ... + P(X = 14)
We can calculate each individual probability using the binomial probability formula as explained earlier. However, this can be quite time-consuming for fourteen different probabilities. Alternatively, we can find the complement of the event "less than three patients having the mild side effect":
P(at least three) = 1 - P(X < 3)
To find P(X < 3), we can use the cumulative probability formula:
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
Now, we already know the probability of having exactly two patients with the mild side effect (P(X = 2)), as calculated in part (A). We just need to calculate P(X = 0) and P(X = 1) to find P(X < 3).
P(X = 0) = (14 C 0) * 0.10^0 * (1-0.10)^(14-0)
= 1 * 1 * (0.9)^14
≈ 0.2281
P(X = 1) = (14 C 1) * 0.10^1 * (1-0.10)^(14-1)
= 14 * 0.10 * (0.9)^13
≈ 0.3560
Now, substitute these values into the cumulative probability formula:
P(at least three) = 1 - P(X < 3)
= 1 - (P(X = 0) + P(X = 1) + P(X = 2))
= 1 - (0.2281 + 0.3560 + 0.2413)
≈ 1 - 0.8254
≈ 0.1746
Therefore, the probability of at least three patients having the mild side effect is approximately 0.1746.