\binom{6}{2} (0.20)^2 (0.80)^4 = 15 \cdot 0.04 \cdot 0.4096 = 0.24576

["# Understanding Binomial Probability: Calculating ( \binom{6}{2} (0.2)^2 (0.8)^4 = 0.24576 )", "In the world of probability, binomial calculations play a crucial role in predicting outcomes in experiments with binary results—think coin flips, success/failure scenarios, or statistical testing. One powerful formula used in this domain is the binomial probability formula:", "[\nP(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}\n]", "This equation helps determine the probability of achieving exactly ( k ) successes in ( n ) independent trials, where the probability of success in a single trial is ( p ).", "---", "## Solving ( \binom{6}{2} (0.2)^2 (0.8)^4 )", "Let’s break down the expression ( \binom{6}{2} (0.20)^2 (0.80)^4 ) step by step to understand how it evaluates to 0.24576.", "### Step 1: Calculate the Binomial Coefficient ( \binom{6}{2} )", "The binomial coefficient ( \binom{n}{k} ), read as "n choose k," counts the number of ways to choose ( k ) successes out of ( n ) trials:", "[\n\binom{6}{2} = \frac{6!}{2!(6 - 2)!} = \frac{6 \ imes 5}{2 \ imes 1} = 15\n]", "So, ( \binom{6}{2} = 15 ).", "### Step 2: Compute ( (0.20)^2 )", "Raise the probability of success to the power of the number of successes:", "[\n(0.20)^2 = 0.04\n]", "### Step 3: Compute ( (0.80)^4 )", "Raise the probability of failure (here, ( 1 - p = 0.80 )) to the power of the number of failures, which is ( n - k = 6 - 2 = 4 ):", "[\n(0.80)^4 = 0.4096\n]", "Why?\nCalculating step-by-step:\n( 0.8^2 = 0.64 ), then ( 0.8^4 = (0.8^2)^2 = 0.64^2 = 0.4096 ).", "---", "### Step 4: Combine All Components", "Now multiply all three components together:", "[\n\binom{6}{2} \cdot (0.20)^2 \cdot (0.80)^4 = 15 \cdot 0.04 \cdot 0.4096\n]", "First:\n( 15 \ imes 0.04 = 0.6 )\nThen:\n( 0.6 \ imes 0.4096 = 0.24576 )", "---", "## Final Result and Real-World Meaning", "[\n\boxed{ \binom{6}{2} (0.20)^2 (0.80)^4 = 0.24576 }\n]", "This means: in 6 independent trials where each has a 20% chance of "success" and an 80% chance of "failure," the probability of exactly 2 successes is 24.576%.", "Binomial distributions like this one are fundamental in statistics, machine learning, quality control, and many areas where event probabilities matter. Mastering this formula helps in accurately modeling real-life scenarios involving independent binary outcomes.", "---", "## Key Takeaways", "- The binomial coefficient ( \binom{6}{2} = 15 ) counts combinations.\n- Raising probabilities leaves the power unchanged and scales each trial independently.\n- Multiplying gives the exact probability for exactly ( k ) successes in ( n ) trials.\n- This calculation applies across fields—from insurance models to genetic probability analysis.", "If you're exploring statistical modeling or probability, understanding this formula is a critical step forward."]