Problem 35 - Algebra and Trigonometry

Question

Effectiveness of a Drug \(A\) certain disease has a mortality rate of 60\(\%\) . A new drug is tested for its effectiveness against this disease. Ten patients are given the drug, and eight of them recover. (a) Find the probability that eight or more of the patients would have recovered without the drug. (b) Does the drug appear to be effective? (Consider the drug effective if the probability in part (a) is 0.05 or less.)

Step-by-Step Solution

Answer

The drug is effective because \( P(X \geq 8) \leq 0.05 \).

Step 1: Define the Binomial Experiment

We know that the probability of recovery without the drug is 0.4 (since the mortality rate is 60%). Let the random variable \( X \) be the number of patients who recover. \( X \) follows a binomial distribution with parameters \( n = 10 \) (the number of trials/patients) and \( p = 0.4 \) (probability of recovery without the drug). We need to find \( P(X \geq 8) \).

Step 2: Calculate the Probability Using Complementary Events

To find \( P(X \geq 8) \), it is easier to use the complement rule: \( P(X \geq 8) = 1 - P(X \leq 7) \). We will calculate \( P(X \leq 7) \) by summing the probabilities for 0 through 7 recoveries.

Step 3: Use Binomial Formula to Find Probabilities

The binomial formula is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \) for \( k = 0, 1, 2, \ldots, n \). Compute \( P(X = k) \) for \( k = 0, 1, 2, 3, 4, 5, 6, 7 \) using \( n = 10 \) and \( p = 0.4 \).

Step 4: Calculate \( P(X \leq 7) \)

Sum up all the probabilities for 0 to 7 recoveries: \[P(X \leq 7) = \sum_{k=0}^{7} \binom{10}{k} (0.4)^k (0.6)^{10-k} \].

Step 5: Compute \( P(X \geq 8) \)

Now, compute \( P(X \geq 8) \) as:\[P(X \geq 8) = 1 - P(X \leq 7)\].

Step 6: Decision on Drug Effectiveness

Compare \( P(X \geq 8) \) to 0.05. If \( P(X \geq 8) \leq 0.05 \), the drug is effective.

Key Concepts

Probability of Recovery

In medical trials, like the one described, we often deal with probabilities related to patient outcomes. Here, we look at the **probability of recovery** from a disease. Without any intervention, the recovery probability is determined by the mortality rate. If a disease has a mortality rate of 60%, this means only 40% of people recover on their own. This probability is vital for understanding how a new treatment might change outcomes. The initial problem requires calculating the probability that eight or more out of ten patients recover without receiving the drug. This involves understanding how these probabilities behave in a binomial setting, which further leads to statistical testing and drawing conclusions about drug effectiveness.

Drug Effectiveness

Drug trials aim to assess whether a new treatment works. The term **drug effectiveness** refers to the ability of a drug to produce a desired result, in this case, increasing the number of recoveries from the disease. To determine effectiveness, we examine the likelihood that such results could occur by chance without the drug. If, without the drug, there's a very low probability of achieving a similar recovery rate, then the drug can be deemed effective. The specific criterion in the problem is that the drug is considered effective if the probability is 5% or lower. This criterion ensures that any observed difference in recovery rate when using the drug is statistically significant and not due to random chance.

Complement Rule

When solving probability problems, sometimes calculating probabilities directly can be complex. The **complement rule** simplifies this by focusing on what doesn't happen instead of what does. For the problem at hand, we're interested in the probability that eight or more patients recover without the drug. Calculating this directly involves numerous computations of binomial probabilities. By using the complement rule, we look at its opposite: fewer than eight recoveries. The formula used is \[ P(X \geq 8) = 1 - P(X \leq 7) \]. In essence, we're subtracting the probability of seven or fewer recoveries from 1. This rule is an invaluable technique in probability, especially within binomial distributions, to save time and reduce calculation complexity.