Big babies: A public health organization reports that \( 30 \% \) of baby boys \( 6-8 \) months old in the United States weigh more than 20 pounds. A sample of 10 babies is studied. Round the answers to three decimal places. Part 1 of 4 (a) What is the probability that exactly 3 of them weigh more than 20 pounds? The probability that exactly 3 of them weigh more than 20 pounds is 0.267.

To find the probability that exactly 3 out of 10 babies weigh more than 20 pounds, we can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where: - \( n = 10 \) (the number of trials) - \( k = 3 \) (the number of successes, or babies weighing more than 20 pounds) - \( p = 0.30 \) (the probability of success on each trial) Calculating it gives us: \[ P(X = 3) = \binom{10}{3} (0.30)^3 (0.70)^{7} \] \[ = \frac{10!}{3!(10-3)!} (0.30)^3 (0.70)^7 \] \[ = 120 \cdot 0.027 \cdot 0.0823543 \approx 0.267 \] Part 1 is confirmed with the probability that exactly 3 of them weigh more than 20 pounds being 0.267. --- Now, let's dive into some extra fun facts about statistics! Did you know that the binomial distribution is often used in real-world settings, such as quality control and game design? It helps manufacturers determine the probability that a certain number of goods will meet quality standards, making it essential for producing products we use daily! Furthermore, if you want to dive deeper into statistics, consider picking up "The Cartoon Guide to Statistics" by Larry Gonick and Woollcott Smith. It's a lively and approachable read that breaks down complex concepts using humor and engaging illustrations, making statistics less intimidating and more enjoyable!