Final answer:
The question relates to the application of probability for a successful basketball free throw by a player and using logarithmic functions to model skill improvement over time. It also touches on the concept of regression and the relationship between exponential and logarithmic functions in mathematics.
Step-by-step explanation:
Understanding Free Throws Probability and Exponential Functions in Basketball
The question relates to the use of mathematics, specifically probability and logarithmic functions, to model the performance in basketball free throws. The situation with Helen, who has a 75 percent success rate with free throws, represents an application of basic probability. With two attempts, we calculate the chance of making the first shot, denoted as event C. If she indeed has a 75% chance of making each shot, then the probability of event C is simply 0.75.
Regarding the function f(x) = 1 + 5.5 * ln(x + 1), this is an example of an exponential relationship expressed with a natural logarithm (ln). This function could model growth phenomena like skill improvement over consecutive days of practice. Inversely, the exponential and natural logarithm functions (ln) can undo each other, which is useful in different areas of mathematics and science, such as calculating the trajectory of a basketball shot or understanding population growth. In these cases, we can express a number like 2 as e raised to the ln of 2, and this technique can be applied to concepts involving doubling or growth rates.
Finally, the concept of a line of best fit using regression is also linked with logarithmic functions. When dealing with experimental data, we often use a statistical process called regression to estimate this line, and the natural log function plays a critical role when analyzing the relationship between variables that grow at a certain rate.