5Question: A zoologist is tracking 5 red birds, 7 green birds, and 3 blue birds in a forest. If she randomly selects 4 birds, what is the probability that she selects at least one bird of each color?

["Title: Understanding Probability: How to Calculate the Chance of Selecting Birds of Every Color – A Zoologist’s Study", "Meta Description:\nExplore the probability of selecting at least one red, green, and blue bird when choosing 4 birds at random from a group of 5 red, 7 green, and 3 blue birds in the wild.", "---", "Abstract\nWhen studying animal behavior or biodiversity in natural settings, understanding random sampling probabilities is crucial. In this article, we examine a practical scenario involving 5 red birds, 7 green birds, and 3 blue birds, totaling 15 birds. By calculating the probability of selecting at least one bird of each color when randomly choosing 4 birds, we uncover key principles of combinatorics and probability essential for wildlife research.", "---", "### Introduction", "Zoologists often conduct field studies involving random sampling of animal populations to draw meaningful ecological insights. A common question arises when researchers select individuals from a mixed group: what is the likelihood of capturing diversity in their sample? This article explores such a scenario using a straightforward but powerful example: tracking 5 red birds, 7 green birds, and 3 blue birds in a forest. By randomly selecting 4 birds, what is the probability they represent at least one of each color?", "---", "### The Problem Explained", "Imagine a forest where a zoologist monitors three distinct bird groups:", "- 5 red birds\n- 7 green birds\n- 3 blue birds\nTotal: 15 birds", "She randomly captures 4 birds without replacement. We seek the probability that her selection includes at least one bird of each color.", "---", "### Step 1: Total Possible Selections", "First, calculate how many ways she can choose any 4 birds from 15:", "[\n\ ext{Total combinations} = \binom{15}{4} = \frac{15!}{4!(15-4)!} = 1365\n]", "---", "### Step 2: Favorable Outcomes — At Least One Bird of Each Color", "To have at least one red, one green, and one blue bird among 4 selected, the distribution of colors must be one of the following combinations:", "1. 2 red, 1 green, 1 blue\n2. 1 red, 2 green, 1 blue\n3. 1 red, 1 green, 2 blue", "We calculate favorable outcomes for each case and sum them.", "---", "#### Case 1: 2 Red, 1 Green, 1 Blue", "[\n\binom{5}{2} \ imes \binom{7}{1} \ imes \binom{3}{1} = 10 \ imes 7 \ imes 3 = 210\n]", "#### Case 2: 1 Red, 2 Green, 1 Blue", "[\n\binom{5}{1} \ imes \binom{7}{2} \ imes \binom{3}{1} = 5 \ imes 21 \ imes 3 = 315\n]", "#### Case 3: 1 Red, 1 Green, 2 Blue", "[\n\binom{5}{1} \ imes \binom{7}{1} \ imes \binom{3}{2} = 5 \ imes 7 \ imes 3 = 105\n]", "---", "### Step 3: Sum Favorable Outcomes", "[\n210 + 315 + 105 = 630\n]", "---", "### Step 4: Calculate the Probability", "[\n\ ext{Probability} = \frac{\ ext{Favorable outcomes}}{\ ext{Total outcomes}} = \frac{630}{1365}\n]", "Simplify by dividing numerator and denominator by 15:", "[\n\frac{630 \div 15}{1365 \div 15} = \frac{42}{91}\n]", "Further simplify by dividing numerator and denominator by 7:", "[\n\frac{6}{13}\n]", "---", "### Final Answer", "The probability that the zoologist selects at least one bird of each color when randomly choosing 4 birds from this forest population is:", "[\n\boxed{\frac{6}{13}}\n]", "---", "### Why This Matters in Wildlife Research", "This probability model helps zoologists understand sampling bias and confidence in observed diversity. Accurately estimating such probabilities ensures reliable data interpretation when working with incomplete or random samples in natural habitats.", "---", "Keywords: probability, zoologist, red birds, green birds, blue birds, random sampling, combinatorics, wildlife research, forest ecology, statistical analysis, biodiversity sampling", "Related Search Terms:\n- probability of selecting mixed color birds\n- calculating diversity in wildlife sampling\n- how many birds to get all colors in a sample\n- statistical methods in animal behavior studies"]