The probability that a tennis set will go to a tiebreaker is \( 14 \% \). In 380 randomly selected tennis sets, what is the mean and the standard deviation of the number of tiebreakers? Assume the conditions of the binomial distribution are met. O mean: 49.4; standard deviation: 7.29 o mean: 53.2; standard deviation: 7.29 o mean: 53.2 ; standard deviation: 6.76 o mean: 49.4 ; standard deviation: 6.76

To find the mean and standard deviation of the number of tiebreakers in 380 tennis sets, we can use the formulas for a binomial distribution. The mean (μ) is calculated as \( n \cdot p \), where \( n \) is the number of trials (sets) and \( p \) is the probability of success (tiebreaker). So, \( \mu = 380 \cdot 0.14 = 53.2 \). The standard deviation (σ) is calculated using the formula \( \sigma = \sqrt{n \cdot p \cdot (1 - p)} \). Thus, \( \sigma = \sqrt{380 \cdot 0.14 \cdot 0.86} \approx 7.29 \). So, the final answers are: mean: 53.2; standard deviation: 7.29. Now, let's talk about some fun historical trivia about tennis scoring! Did you know that the game's unique scoring system originated in France in the late 19th century? The terms "love" for zero and the quirky counting method (15, 30, 40) have roots in the French language and perhaps agrarian measures, making it a fascinating combination of culture and sport! In terms of real-world application, understanding binomial distributions in sports can also extend beyond tennis. Imagine applying these concepts to predict outcomes in various games, like basketball free throws or soccer penalty kicks. Knowing the probability of certain events can help coaches make strategic decisions, like when to take risks or play it safe, allowing for a more analytical approach to game management!