Testing a Drug A drug manufacturer claims that a drug cures a rare skin disease 75% of the time. The claim is checked by testing the drug on 100 patients. If at least 70 patients are cured, then this claim will be accepted. Use this information in Exercises 31 and 32.


Find the probability that the claim will be accepted, assuming that the actual probability that the drug cures the skin disease is 65%.

Verified step by step guidance

1. Step 1: Identify the type of probability distribution. Since the problem involves a fixed number of trials (100 patients), each with two possible outcomes (cured or not cured), and a constant probability of success (65%), this is a binomial distribution. The binomial distribution is defined as P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.
2. Step 2: Define the random variable and parameters. Let X represent the number of patients cured. Here, n = 100 (number of trials), p = 0.65 (probability of success), and we are interested in the probability that at least 70 patients are cured, i.e., P(X ≥ 70).
3. Step 3: Rewrite the probability using the complement rule. Since calculating P(X ≥ 70) directly can be cumbersome, use the complement rule: P(X ≥ 70) = 1 - P(X < 70). This simplifies to P(X ≥ 70) = 1 - P(X ≤ 69).
4. Step 4: Use the cumulative distribution function (CDF) of the binomial distribution. The CDF gives the probability of X being less than or equal to a certain value. Calculate P(X ≤ 69) using the binomial CDF formula or a statistical software/calculator. The formula for the CDF is the sum of binomial probabilities: P(X ≤ 69) = Σ (from k=0 to 69) [(n choose k) * p^k * (1-p)^(n-k)].
5. Step 5: Subtract the CDF value from 1 to find the final probability. Once you have P(X ≤ 69), subtract it from 1 to get P(X ≥ 70). This result represents the probability that the claim will be accepted, assuming the actual probability of curing the disease is 65%.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this scenario, curing patients can be seen as a success, and the distribution helps calculate the probability of curing a certain number of patients out of 100, given a specific success rate.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. In this case, the null hypothesis would be that the drug cures 65% of patients, while the alternative hypothesis is that it cures at least 70 patients. The test evaluates whether the observed data provides sufficient evidence to reject the null hypothesis.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value. In this context, it involves calculating the probability of curing at least 70 patients out of 100, which requires summing the probabilities of curing 70, 71, ..., up to 100 patients using the binomial distribution with a success rate of 65%.