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Chapter 4: Problem 45

An experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a \(10-\mathrm{mg}\) dose of the drug is lethal \(5 \%\) of the time within the first 4 hours; of the animals alive at 4 hours, \(10 \%\) will die in the next 4 hours. What is the probability that 1 rat will die in the 8-hour period?

Short Answer

Expert verified
The probability that exactly 1 rat will die in the 8-hour period is approximately 0.1479.

Step by step solution

01

Calculate Probability of Death in First 4 Hours

For a single rat, the probability of dying within the first 4 hours due to the drug is given as 5%, or \( P_1 = 0.05 \).
02

Calculate Probability of Survival in First 4 Hours

Using the complementary probability, the probability of the rat surviving the first 4 hours is \( P_{1s} = 1 - 0.05 = 0.95 \).
03

Calculate Probability of Death in Next 4 Hours

Given that the rat survived the first 4 hours, the probability it dies in the next 4 hours is 10%, or \( P_2 = 0.10 \).
04

Calculate Overall Probability of Dying

The probability of a rat dying in the total 8-hour period is the sum of the probabilities of: dying in the first 4 hours and surviving then dying in the next 4 hours. This is calculated as: \( P_{total} = P_1 + (P_{1s} \times P_2) = 0.05 + (0.95 \times 0.10) = 0.145 \).
05

Apply Binomial Distribution for 1 Death

We use the binomial probability formula since we want exactly 1 rat to die out of 20. The formula is \( P(X = 1) = \binom{n}{k}p^k(1-p)^{n-k} \), where \( n = 20 \), \( k = 1 \), and \( p = 0.145 \). This gives \( P(X = 1) = \binom{20}{1} (0.145)^1 (1-0.145)^{19} \).
06

Calculate the Binomial Probability

Calculate \( \binom{20}{1} = 20 \), and then \( (0.145)^1 = 0.145 \), and \( (1-0.145)^{19} \approx 0.0510 \). Thus, \( P(X = 1) = 20 \times 0.145 \times 0.0510 = 0.1479 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Distribution
The binomial distribution is a fundamental statistical method used to model the number of successes in a fixed number of independent Bernoulli trials. Each trial has only two possible outcomes: success or failure.
  • In the context of this exercise, a 'success' is defined as the death of a rat within the 8-hour test period.
  • The probability of 'success' (a rat dying) is calculated in multiple steps, leading to an overall probability of 0.145.
Now, we need to determine the probability of exactly one success (one rat dying) out of 20 trials (20 rats). For this, we apply the binomial probability formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]In which:
  • \(n\) is the total number of trials (rats), which is 20.
  • \(k\) is the desired number of successes (rats dying), which is 1.
  • \(p\) is the probability of success in each trial (0.145, as calculated in the solution).
By substituting these values into the formula, we calculate the chance of exactly one rat dying as approximately 0.1479.
Drug Potency Testing
Drug potency testing in the sphere of biostatistics involves assessing the efficacy and safety of a pharmaceutical product. In this exercise, we focus on testing a drug's lethality on a specific population (rats).
  • The primary aim is to determine how the drug affects the subjects over a set period鈥攊n this case, 8 hours.
  • Biostatistical methods help us quantify these effects and predict outcomes based on probability models.
For a drug like the one tested here, part of the evaluation involves calculating the likelihood of a lethal outcome for the subjects over different time frames (first 4 hours vs. the subsequent 4 hours). By combining these probabilities, we hope to better understand the drug's effects. When performing these assessments:
  • Statistical tools like the binomial distribution offer a structured way to predict the impact on a larger population.
  • This methodology is crucial, especially when transitioning findings from animal models to potential human implications.
Survival Analysis
Survival analysis is a set of statistical techniques focusing on the expected duration until one or more events happen, such as death in biological organisms. This is ideally suited to studies like the one presented in the exercise, which investigates survival outcomes across time periods. The method uses probabilities to anticipate when particular 'events' (in this case, the death of rats) might occur and understands the factors affecting these timelines.
  • First, we estimate the probability of surviving through different time intervals.
  • In this scenario, the probability was calculated for the rats surviving the first 4 hours as well as after those initial 4 hours until the 8-hour mark.
The information gleaned from survival analysis plays a key role in drug testing by:
  • Determining effective dosages that minimize risk.
  • Predicting the survival rates of the population under study, thus informing future research directions or clinical trials.
Overall, survival analysis empowers researchers to make informed decisions on the viability and safety of new medicinal treatments.

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Most popular questions from this chapter

The two-stage model of carcinogenesis is based on the premise that for a cancer to develop, a normal cell must first undergo a "first hit" and mutate to become a susceptible or intermediate cell. An intermediate cell then must undergo a "second hit" and mutate to become a malignant cell. A cancer develops if at least one cell becomes a malignant cell. This model has been applied to the development of breast cancer in females (Moolgavkar et al. \([15])\) Suppose there are \(10^{8}\) normal breast cells and 0 intermediate or malignant breast cells among 20 -year-old females. The probability that a normal breast cell will mutate to become an intermediate cell is \(10^{-7}\) per year. What is the expected number of intermediate cells by age 45?

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What is the probability of obtaining at least 6 events for a Poisson distribution with parameter \(H=4.0 ?\)
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