Solution: The probability follows a binomial distribution with parameters $ n = 5 $ and $ p = \frac{1}{3} $. The probability of exactly 2 successes is $ \binom{5}{2} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^3 $. Calculating:

["Solution: Probability Follows a Binomial Distribution with Parameters ( n = 5 ) and ( p = \frac{1}{3} )", "When analyzing experiments with a fixed number of independent trials and a constant success probability, the binomial distribution provides a powerful model. In this article, we explore how to calculate the probability of exactly 2 successes in 5 trials, where each trial has a success probability of ( \frac{1}{3} ).", "The binomial distribution describes the probability of achieving exactly ( k ) successes in ( n ) independent Bernoulli trials. Its formula is:", "[\nP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\n]", "Where:\n- ( n = 5 ) (number of trials),\n- ( p = \frac{1}{3} ) (probability of success on each trial),\n- ( k = 2 ) (exact number of successes we want to calculate).", "### Step-by-step Calculation", "1. Identify the binomial parameters:\n ( n = 5 ), ( k = 2 ), ( p = \frac{1}{3} ), and therefore ( 1 - p = \frac{2}{3} ).", "2. Compute the binomial coefficient:\n [\n \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \ imes 4}{2 \ imes 1} = 10\n ]", "3. Calculate the probability components:\n - Success probability raised to the power of ( k ):\n [\n p^k = \left(\frac{1}{3}\right)^2 = \frac{1}{9}\n ]\n - Failure probability raised to the power of ( n - k ):\n [\n (1-p)^{n-k} = \left(\frac{2}{3}\right)^3 = \frac{8}{27}\n ]", "4. Combine all parts into the full formula:\n [\n P(X = 2) = \binom{5}{2} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^3 = 10 \ imes \frac{1}{9} \ imes \frac{8}{27}\n ]", "5. Multiply step by step:\n First, ( 10 \ imes \frac{1}{9} = \frac{10}{9} )\n Then, ( \frac{10}{9} \ imes \frac{8}{27} = \frac{80}{243} )", "### Final Result", "Thus, the probability of exactly 2 successes in 5 trials with success probability ( \frac{1}{3} ) is:", "[\n\boxed{\frac{80}{243}}\n]", "This result reflects how the binomial framework efficiently captures the likelihoods in discrete probability settings—ideal for applications in quality control, survey sampling, and clinical trials. By understanding the binomial distribution and applying its formula correctly, complex probabilistic outcomes become clear and calculable."]