# Percentages of success in basketball

• MHB
• Yankel
In summary, the basketball player's success rate is not necessarily 80% at any given time, but it is always greater than or equal to 80%.
Yankel
Hello all,

I wasn't sure if this question should be placed here or on algebra forum, but it is related to probability, so I put it here.

A basketball player shoots several times during some time period in a game, and scores less than 80% of his shots (let's say after the first quarter, but the time doesn't really matters). At the end of the game, he scored more than 80% of his shots. Is it necessary that at some point during the game, his success was exactly 80% ?

My intuition say that the answer is no, because this is not a continuous function, but every example I tried setting up, showed that 80% was achieved. What do you think ?

Yankel said:
Hello all,

I wasn't sure if this question should be placed here or on algebra forum, but it is related to probability, so I put it here.

A basketball player shoots several times during some time period in a game, and scores less than 80% of his shots (let's say after the first quarter, but the time doesn't really matters). At the end of the game, he scored more than 80% of his shots. Is it necessary that at some point during the game, his success was exactly 80% ?

My intuition say that the answer is no, because this is not a continuous function, but every example I tried setting up, showed that 80% was achieved. What do you think ?

Your correct that the answer is no. In terms of finding an example to show this the easiest way is to find some fraction that is greater than and not equal to 80% where subtracting one from the numerator and one from the denominator is less than and not equal to 80%. In other words find a fraction such that making the last shot pushes them over 80% success and that before making the last shot they had less than 80% success.

Find an x,y such that
$$\displaystyle \frac{x}{y} > 80%$$ and
$$\displaystyle \frac{x-1}{y-1} < 80%$$

Actually, I think the claim is true. First, I tried some elementary examples and they appeared to be true so my intuition was that the claim should be true. Suppose that he shots $n$ times during the game, then each time he shots we can compute his succes rate. The game starts and he shots ... he scores, then his succes rate at that time is $1/1 = 100\%$. Suppose he shots again after a few minutes but misses then his succes rate is now $1/2 = 50\%$. So in fact, you can represent the game as a finite sequence
$$\left(\frac{a_i}{i}\right)_{i=1}^{n},$$
where the $a_i's$ are increasing by one but not necessarily each consecutive term.

Statement
The question can then be reformulated as: given that $a_n/n > (4/5)$ and at some point in the game $a_j/j < (4/5)$ for $j<n$. Does this imply that there exists an index $k$ such that $a_k /k = 0.8$ where $j<k<n$?

Proof
​The answer is yes. We have given $\displaystyle \frac{a_n}{n} > \frac{4}{5} > \frac{a_j}{j}$. Hence, $5a_n>4n$ and $5a_j < 4j$. Let $b_k=5a_k-4k$. As $a_{k+1}$ is either $a_k$ or $a_k+1$, we have $$b_{k+1}=5a_{k+1}-4k-4=\begin{cases}5a_k-4k-4&=b_k-4\quad \text{ or}\\5(a_k+1)-4k-4&=b_k+1\end{cases}$$Thus if the integer $b_k$ increases, it only increases in steps of $1$. From $b_j<0<b_n$ we see that there must be a ​first index $k$ between $j$ and $n$ for which $b_k\ge 0$. Then $b_k\le b_{k-1}+1$ and $b_{k-1}<0$, hence $b_k=0$ as desired.

Please, correct me if I am wrong.

## 1. What is the average percentage of successful shots in basketball?

The average percentage of successful shots in basketball varies depending on the level of play. In professional basketball, the average field goal percentage is around 45%, while in college basketball it is around 43%. However, this percentage can also be affected by individual player skill and the specific team's strategy.

## 2. How do percentages of success in basketball differ between men and women?

In general, men tend to have a higher percentage of successful shots in basketball compared to women. This is due to a number of factors such as physical capabilities, training, and overall competitiveness. However, it is important to note that there are highly skilled and successful female basketball players who have comparable percentages to their male counterparts.

## 3. What is the correlation between a player's shooting percentage and their overall success in basketball?

A player's shooting percentage can be a good indicator of their overall success in basketball, but it is not the only factor that determines success. Other important factors include teamwork, defense, and overall basketball IQ. Additionally, a player's shooting percentage can fluctuate depending on various factors such as matchups, injuries, and fatigue.

## 4. How do percentages of success in basketball differ between different positions?

Percentages of success in basketball can vary between different positions on the court. For example, point guards tend to have a higher percentage of successful free throws due to their ball-handling and passing skills, while centers tend to have a higher percentage of successful field goals due to their size and proximity to the basket. However, this can also vary depending on the individual player's skill and style of play.

## 5. Can percentages of success in basketball be improved through training?

Yes, percentages of success in basketball can be improved through training. Shooting drills and practice can help improve a player's shooting percentage, while strength and conditioning training can also improve overall performance on the court. Additionally, working on mental aspects such as focus and confidence can also positively impact a player's percentages of success in basketball.

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