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Chapter 3: Problem 60

A particular concentration of a chemical found in polluted water has been found to be lethal to \(20 \%\) of the fish that are exposed to the concentration for 24 hours. Twenty fish are placed in a tank containing this concentration of chemical in water. a. Find the probability that exactly 14 survive. b. Find the probability that at least 10 survive. c. Find the probability that at most 16 survive. d. Find the mean and variance of the number that survive.

Short Answer

Expert verified
a. Use Binomial formula; b. Use complement; c. Add probabilities; d. Mean = 16, variance = 3.2.

Step by step solution

01

Identify the Distribution

This problem involves a fixed number of trials (20 fish), each with two possible outcomes (survive or not), with constant probability of success (surviving), typical of a Binomial distribution. Each fish has a probability of survival of \(1 - 0.2 = 0.8\). Thus, we have a Binomial distribution with \(n = 20\) trials and probability of success \(p = 0.8\).
02

Calculate the Probability of Exactly 14 Survivors

Using the Binomial probability formula \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\), calculate the probability that exactly 14 fish survive: \[ P(X = 14) = \binom{20}{14} (0.8)^{14} (0.2)^6 \]. Compute this using a calculator to get the probability.
03

Calculate the Probability of At Least 10 Survivors

To find \(P(X \geq 10)\), we use the complement rule: \(P(X \geq 10) = 1 - P(X < 10)\). Calculate \(P(X < 10)\) by adding the probabilities \(P(X = 0), P(X = 1), ..., P(X = 9)\) using the Binomial formula and subtract from 1.
04

Calculate the Probability of At Most 16 Survivors

To find \(P(X \leq 16)\), add the probabilities \(P(X = 0), P(X = 1), ..., P(X = 16)\) using the Binomial formula. Alternatively, subtract \(P(X = 17), P(X = 18), P(X = 19), P(X = 20)\) from 1.
05

Calculate the Mean and Variance of Survivors

For a Binomial distribution, the mean \(\mu\) is given by \(np\) and the variance \(\sigma^2\) is given by \(np(1-p)\). Using these formulas, calculate the mean and variance: \(\mu = 20 \times 0.8 = 16\) and \(\sigma^2 = 20 \times 0.8 \times 0.2 = 3.2\).

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Key Concepts

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

Probability
Probability is central to understanding events' likelihoods and outcomes. In this exercise, we focus on fish surviving in polluted water. Here, each fish's fate is an independent event. The probability of each fish surviving is 80% (or 0.8), as given.
  • The complement of an event is also useful, as it allows us to understand the probability of not happening, here 20% (or 0.2) for not surviving.
  • Probabilities help us predict how many fish survive with a given chemical concentration over 24 hours.
Through the Binomial formula, we calculate exact survival numbers, like figuring out how likely exactly 14 fish survive out of 20. The framework of Probability acts as the foundation upon which these calculations are based.
Mean and Variance
Mean and variance are key concepts in statistics, essential for understanding the average outcomes and the data spread, respectively. In this problem set, they help gauge the overall surviving fish and any variability.
  • The mean, also called the expected value, in a Binomial distribution is calculated with the formula \(np\). For our scenario, it's \(20 \times 0.8 = 16\). This average tells us we'd expect 16 fish survive on average.
  • The variance measures the distribution's spread, given by \(np(1-p)\). Here it's \(20 \times 0.8 \times 0.2 = 3.2\). Variance indicates the likelihood of the number of surviving fish varying around the mean of 16.
While the mean provides a central, expected value, variance outlines the expected deviation, offering a fuller picture of probability outcomes.
Probability Distribution
Probability distribution in this context directly relates to our understanding of how survival chances are spread among the 20 fish. A Binomial Distribution is a type of probability distribution. It applies here because we're observing a series of trials with only two outcomes (survive or not), known probabilities and independent trials.
  • In this Binomial setting, we're using 20 trials, each having a success probability of 0.8 for fish survival.
  • The distribution shows us every possible outcome (number of survivors), along with their likelihood, from zero survivors to all 20 surviving.
Hence, calculating probabilities like exactly 14 survivors or at least 10 surviving utilizes Binomial Distribution, facilitating precise modeling of outcomes, despite random potential variations.

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Most popular questions from this chapter

Of the volunteers donating blood in a clinic, \(80 \%\) have the Rhesus (Rh) factor present in their blood. a. If five volunteers are randomly selected, what is the probability that at least one does not have the Rh factor? b. If five volunteers are randomly selected, what is the probability that at most four have the Rh factor? c. What is the smallest number of volunteers who must be selected if we want to be at least \(90 \%\) certain that we obtain at least five donors with the Rh factor?

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