Current learning objective: Notation for the Binomial: B = Binomial Probability Distribution Function Question 13 Score: 0 of 4 points According to the American Red Cross, \( 10.4 \% \) of all Connecticut residents have Type B blood. A random sample of 23 Connecticut residents is taken. \( \begin{array}{ll}\text { X } & \text { the number of CT residents that have Type B blood, of the } 23 \text { sampled. What is the expected value of the random variable } X \text { ? } \\ \text { A } 2.392 \\ \text { B } 2.576 \\ \text { C } 2.415 \\ \text { D } 2.438 \\ \text { E } 1.978\end{array} \)

To find the expected value \( E(X) \) for a binomial distribution, you can use the formula \( E(X) = n \times p \), where \( n \) is the number of trials (23 residents) and \( p \) is the probability of success (0.104 for Type B blood). Plugging in the values, you get: \[ E(X) = 23 \times 0.104 \approx 2.392 \] So, the expected value of the random variable \( X \) is approximately 2.392, which corresponds to option A! Calculating expected values not only gives us insights in this situation but also helps in various fields such as quality control in manufacturing or predicting outcomes in healthcare. It’s a handy tool to manage risks and expectations. Understanding this concept helps demystify how probabilities play out in real life—making it impactful for everything from elections to product launches! Moreover, many people mistakenly think that probability refers to a single outcome when it’s quite the opposite! Make sure to visualize distributions and grasp that these values represent averages over many trials, not certainties in individual trials. Keep practicing with real-world examples, and you'll get the hang of it in no time!