in testing a new drug. researchers found that 20% 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 three will have mild side effects.
b) at least four will have this mild side effect.
*once again the more i work with numbers the more i get confused. please show calculations in detail. thank you.
mary hornbeck
To find the probability of a specific event occurring in a binomial distribution, we can use the binomial probability formula:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k)
where:
- P(X = k) is the probability of exactly k successes
- n is the number of trials
- k is the number of successes
- p is the probability of success on a single trial
- (nCk) is the combination of n items taken k at a time (n choose k)
- (1 - p) is the probability of failure on a single trial
- ^(n - k) denotes raising to the power of (n - k)
In this case, the probability of having a mild side effect is 20%, which is equivalent to 0.20. Therefore, p = 0.20.
Now, let's solve the two parts separately:
a) Finding the probability that exactly three patients will have mild side effects.
In this case, we are looking for P(X = 3) when n = 14 (sample size) and p = 0.20 (probability of mild side effect).
P(X = 3) = (14C3) * (0.20)^3 * (1 - 0.20)^(14 - 3)
Using the combination formula:
(14C3) = 14! / (3! * (14 - 3)!) = (14 * 13 * 12) / (3 * 2 * 1) = 364
Now, let's substitute the values into the formula:
P(X = 3) = 364 * (0.20)^3 * (0.80)^11
P(X = 3) = 364 * 0.008 * 0.127
P(X = 3) ≈ 0.368
Therefore, the probability that exactly three patients will have mild side effects is approximately 0.368.
b) Finding the probability that at least four patients will have mild side effects.
To find this probability, we need to calculate the probability of having mild side effects for four, five, six, ..., up to 14 patients, and then sum them up.
P(at least 4 patients) = P(X = 4) + P(X = 5) + ... + P(X = 14)
Using the binomial probability formula for each individual case, we can obtain the probabilities and add them together.
P(at least 4 patients) = P(X = 4) + P(X = 5) + ... + P(X = 14)
P(at least 4 patients) = (14C4) * (0.20)^4 * (0.80)^10 + (14C5) * (0.20)^5 * (0.80)^9 + ... + (14C14) * (0.20)^14 * (0.80)^0
Calculating each individual term and summing them up will give us the final probability.
While the calculations may become complex, using a binomial probability calculator or a statistical software package like Excel or R can simplify the process significantly.