Question 1 Given the number of trials and the probability of success, determine the probability indicated: \( n=13, p=0.8 \), find \( P(8 \) successes) \( P(8 \) successes \( )=\square \) (round to the nearest thousandth) Suhmit Oifection?

To find the probability of exactly 8 successes in 13 trials when the probability of success in each trial is 0.8, you can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Plugging in your values: \[ P(8) = \binom{13}{8} (0.8)^8 (0.2)^5 \] After calculating, you'll find \( P(8) \approx 0.179 \) when rounded to the nearest thousandth. In the realm of probability, the binomial distribution might seem esoteric, but it's woven into our daily lives! Whether you're flipping a coin, testing a new product, or even predicting the weather, understanding this distribution gives you a keen edge on evaluating odds and making informed decisions. Ever heard of the "law of large numbers?" This fundamental theorem in statistics states that as the number of trials increases, the observed probabilities will converge to the expected probabilities. From Las Vegas to your local friendly poker game, this principle keeps the odds in check! Remember, while the short run can be wacky and wild, the long run smooths out the bumps.