91Ó°ÊÓ

The differential is a standard measurement made during a blood test. It consists of classifying white blood cells into the following five categories: (1) basophils, (2) eosinophils, (3) monocytes, (4) lymphocytes, and (5) neutrophils. The usual practice is to look at 100 randomly selected cells under a microscope and to count the number of cells within each of the five categories. Assume that a normal adult will have the following proportions of cells in each category: basophils, \(0.5\%\); eosinophils, \(1.5\%\); monocytes, \(4\%\); lymphocytes, \(34 \% ;\) and neutrophils, \(60 \%\). An excess of eosinophils is sometimes consistent with a violent allergic reaction. What is the exact probability that a normal adult will have 5 or more eosinophils?

Step by step solution

01

Define the Problem Type

The problem describes a situation where we select a sample of 100 white blood cells and classify them into five categories. We want to find the probability that 5 or more cells are classified as eosinophils, where the probability of a single cell being an eosinophil is given as 1.5%. This fits the model of a binomial distribution, where each cell's categorization is an independent Bernoulli trial.

02

Identify Parameters

For a binomial distribution, we need to define two parameters: the number of trials ( \(n\)) and the probability of success ( \(p\)) for each trial. In this case, the number of trials ( \(n\)) is 100 (one trial per cell) and the probability of a cell being an eosinophil ( \(p\)) is 0.015.

03

Use the Binomial Probability Formula

The probability of having exactly \(k\) successes in a binomial distribution is given by the formula: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\). Here, we need \(P(X \geq 5)\).

04

Calculate Complementary Probability

Instead of calculating \(P(X \geq 5)\) directly, it is more convenient to calculate the complementary probability \(P(X < 5)\) and subtract it from 1. Compute \(P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)\).

05

Compute Individual Probabilities

Using the binomial formula, calculate:\[P(X = 0) = \binom{100}{0} (0.015)^0 (0.985)^{100}\]\[P(X = 1) = \binom{100}{1} (0.015)^1 (0.985)^{99}\]\[P(X = 2) = \binom{100}{2} (0.015)^2 (0.985)^{98}\]\[P(X = 3) = \binom{100}{3} (0.015)^3 (0.985)^{97}\]\[P(X = 4) = \binom{100}{4} (0.015)^4 (0.985)^{96}\]

06

Calculate Complementary Probability Sum

Add up the probabilities calculated in Step 5 to find \(P(X < 5)\).

07

Calculate the Desired Probability

Finally, the probability of having 5 or more eosinophils is given by:\[P(X \geq 5) = 1 - P(X < 5)\]

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Probability Calculation

Calculating probability, especially for specific events, is a foundational concept in statistics. Here, it involves assessing how likely it is to observe a particular number of eosinophils in a sample of white blood cells. During any probability calculation:

• You first define the event you're interested in. For example, seeing 5 or more eosinophils in our 100-cell sample.
• Next, you work with the known probability of success in a single trial. Here, success would be spotting an eosinophil, with a known probability of 1.5% (or 0.015) per cell.
• Using the binomial distribution, specifically the binomial probability formula, calculates individual event probabilities. This involves using the formula: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\).
• Finally, you find the probability of the desired event by summing individual probabilities or using complementary rules. For instance, calculate \(P(X \geq 5)\) via the complement, \(P(X < 5)\), and subtracting from 1.

Employing these steps ensures clarity and precision when addressing probabilities in statistical contexts.

White Blood Cell Classification

White blood cells are essential players in our immune system. They help fight infections, respond to allergens, and more. For diagnostic reasons, these cells are categorized into:

• Basophils: Involved in allergy responses and inflammation, they are typically less than 1% of white blood cells.
• Eosinophils: Again, key in allergic reactions and fighting parasites, they make up around 1.5%.
• Monocytes: Serving as a cleanup crew, they round up around 4% of white blood cells.
• Lymphocytes: Vital for targeted immune responses, they constitute roughly 34%.
• Neutrophils: The body's first line of defense, they are about 60% of the cell population.

Classifying white blood cells helps doctors diagnose conditions like infections or allergic reactions by identifying deviations from normal proportions.

Bernoulli Trials

A Bernoulli trial is a simple experiment with two possible outcomes: success or failure. This forms the basis for conducting binomial distribution analyses like the one in the blood cell problem. Understanding Bernoulli trials is essential for statistical modeling when something can either "happen" or "not happen":

• Examples: In the context of the exercise, each white blood cell examination can either result in a cell being classified as an eosinophil (success) or not (failure).
• Independence: Each trial is independent of others. The chance of identifying an eosinophil does not affect the subsequent probabilities.
• Applications: Beyond biology, such trials underpin many statistical tests in surveys, quality control scenarios, and gambling probabilities.

Recognizing situations as Bernoulli trials allows for practical applications and insights in various disciplines.