In summary, the conversation discusses the expected number of points resulting from a "one and one" situation in basketball, where a player has the potential to shoot two penalty shots if they land the first shot. The outcome of the second shot is independent of the first. The expected value is calculated using the formula E(x) = S x P(x), and a table is set up to determine the success rate of the shots. The final result is that the expected number of points is 1.19.
Hi, I'm a bit stuck on this problem

A basketball player misses 30% of his free throws. He ends up in a situation where he has the potential to shoot two penalty shots if and only if he lands the first shot (called a one and one, I believe). The outcome of the 2nd shot is independent of the first.

> Find the expected number of points resulting from the "one and one"

Here I'm assuming expected number equates to expected value -> E(x) = S x P(x) (S = sigma)

I've set up a table thusly so that I might take the sum of the 3rd column (x denotes successful shots):
x____P(x)____xP(x)
0...?...0
1.....
2.....

The issue that I am having is how to express the dependence of even taking a second shot upon this first. My gut is telling me that the first entry under P(x) ought to be .3 but after that I'm lost. Any pointers would be appreciated!

Then, 30% of the "one and one" , he misses the first shot, and gets 0 points (prob 30%).
And he lands 70% of the first shot.
Then, he misses the second throw, getting only 1 point (prob 0.7 * 0.3 = 21%).
But he lands 70% of the 2nd throw, getting 2 points (prob 0.7 * 0.7 = 49%).

So, the expected number of points is 1*0.21 + 2*.49 = 1.19 .

Thanks much, I actually managed to figure it out while taking a dinner break from problem sets - you beat me back it seems!

## 1. What is the purpose of conducting independent statistics in basketball?

Independent statistics in basketball are used to objectively measure and analyze the performance of individual players, teams, and the game as a whole. This allows coaches, players, and analysts to identify strengths, weaknesses, and areas for improvement.

## 2. How is independent statistics different from traditional team statistics?

Independent statistics focus on individual performance rather than team performance. Traditional team statistics, such as points scored and rebounds, do not take into account factors such as player usage, efficiency, and impact on the game.

## 3. What types of independent statistics are commonly used in basketball?

Some commonly used independent statistics in basketball include player efficiency rating (PER), true shooting percentage (TS%), and win shares (WS). These statistics take into account various aspects of a player's performance, such as scoring, rebounding, and defense, to provide a comprehensive evaluation.

## 4. How are independent statistics collected and calculated?

Independent statistics are collected through various methods, including manual tracking, video analysis, and advanced statistics software. The data is then compiled and calculated using specific formulas and algorithms to provide meaningful and objective insights.

## 5. Can independent statistics be used to predict the outcome of games?

While independent statistics provide valuable information, they should not be solely relied upon for predicting game outcomes. Other factors, such as team chemistry, injuries, and coaching strategies, also play a significant role in determining the outcome of a game.

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