The probability of surviving a particular disease without treatment is 1/3. Dr. Wort is testing a new drug to treat this disease and finds that 40 out of 100 people who took the drug survived.  What is the probability of obtaining these results if the drug is ineffective? 

This problem can be approached by determining the probability of obtaining the observed result (40 survivors out of 100) under the assumption that the drug is **ineffective**, meaning the probability of survival remains \( \frac{1}{3} \) (the same as without treatment).

We can model this as a **binomial probability problem**:

- **Number of trials (n)**: 100 (patients)
- **Probability of success (p)**: \( \frac{1}{3} \) (survival without the drug)
- **Observed number of successes (k)**: 40 survivors

The probability of exactly \( k \) successes in a binomial distribution is given by:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

Where:
- \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes from \( n \) trials,
- \( p^k \) is the probability of \( k \) successes,
- \( (1 - p)^{n - k} \) is the probability of \( n - k \) failures.

Let's calculate this probability for \( k = 40 \), \( n = 100 \), and \( p = \frac{1}{3} \). I'll compute the result.The probability of obtaining exactly 40 survivors out of 100, assuming the drug is ineffective (with the survival probability of \( \frac{1}{3} \)), is approximately \( 0.0308 \), or 3.08%.

This indicates that there is about a 3.08% chance of observing this outcome if the drug does not have any effect on survival.

Would you like more details or have any questions?

Here are some follow-up questions for further exploration:

1. How does the result change if the survival probability without treatment was 1/2?
2. What if instead of 40, 45 patients survived? How would the probability change?
3. How would you calculate the probability that fewer than 40 patients survived?
4. What is the expected number of survivors if the drug is ineffective?
5. How can we test if the drug is statistically effective based on these results?

**Tip**: The binomial distribution is commonly used for modeling binary outcomes (success/failure) over repeated trials.

The probability of surviving a particular disease without treatment is 1/3. Dr. Wort is testing a new drug to treat this disease and finds that 40 out of 100 people who took the drug survived. What is the probability of obtaining these results if the drug is ineffective?

Calculate the probability of observing 40 survivors out of 100 patients using a binomial distribution. Given that the survival rate without treatment is 1/3, this problem explores the likelihood of the drug being ineffective. Key concepts include binomial distributions and probability theory.

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