A bag contains 5 red, 7 blue, and 8 green marbles. Two marbles are drawn at random without replacement. What is the probability both are blue?
["Calculate the Probability of Drawing Two Blue Marbles Without Replacement", "When analyzing random draws from a finite set, understanding probability distributions helps predict outcomes with precision. In this scenario, we have a marble bag containing marbles of three different colors: 5 red, 7 blue, and 8 green. Our objective is to find the probability that both marbles drawn at random—without replacement—are blue.", "### The Setup", "The bag contains:\n- Red marbles: 5\n- Blue marbles: 7\n- Green marbles: 8", "Total number of marbles = 5 + 7 + 8 = 20 marbles", "### Step-by-Step Probability Calculation", "#### Step 1: Probability of First Blue Marble\nWhen drawing the first marble, the chance it is blue is the ratio of blue marbles to the total:\n[\nP(\ ext{First blue}) = \frac{7}{20}\n]", "#### Step 2: Adjust for Drawing Without Replacement\nAfter removing one blue marble, the bag contains 19 marbles, with 6 blue left. The probability that the second marble is also blue is:\n[\nP(\ ext{Second blue | First blue}) = \frac{6}{19}\n]", "#### Step 3: Multiply Probabilities\nSince both events depend sequentially, multiply the probabilities:\n[\nP(\ ext{Both blue}) = \frac{7}{20} \ imes \frac{6}{19} = \frac{42}{380}\n]", "Simplify the fraction:\n[\n\frac{42}{380} = \frac{21}{190}\n]", "### Final Answer", "The probability of drawing two blue marbles in succession without replacement is (\frac{21}{190}), or approximately 0.1105 (11.05%).", "---", "This example highlights how conditional probability applies in finite sampling without replacement. Knowing the exact counts of each outcome ensures accurate predictions—useful in games, surveys, and statistical modeling. Whether you’re calculating odds, analyzing data, or designing games, understanding such probabilities empowers better decision-making.", "Keywords: probability of two blue marbles, draw without replacement, probability calculation, marbles problem, blue marble probability, conditional probability, finite probability, combinatorics application."]
