Question

Basketball strategy. Late in a basketball game, the team that is behind often fouls someone in an attempt to get the ball back. Usually the opposing player will get to shoot foul shots "one and one," meaning he gets a shot, and then a second shot only if he makes the first one. Suppose the opposing player has made \(72 \%\) of his foul shots this season. Estimate the number of points he will score in a one-and-one situation.

Short answer

The player is expected to score approximately 1.24 points in a one-and-one situation.

Step-by-step solution

Understand the Problem

In a one-and-one free throw situation, the player only gets a second shot if he makes the first one. We need to calculate the expected, or average, number of points the player will score given his foul shot success rate.

Define the Success Rates

The player's success rate for making any given foul shot is 72%, or 0.72. This is the probability that he makes the first shot.

Calculate Expected Points from First Shot

If the player makes the first shot (which happens with probability 0.72), he scores 1 point with certainty and gets a chance for a second shot. The expected points from the first shot is obtained by multiplying the probability of making it by the points gained: \[ E(\text{First Shot}) = 0.72 \times 1 = 0.72 \]

Calculate Expected Points from Second Shot

The second shot is only attempted if the first shot is successful. Thus, the probability of being in a position to make the second shot is 0.72. Given the player's 72% success rate, the expected points from the second shot is:\[ E(\text{Second Shot}) = 0.72 \times 0.72 = 0.5184 \]

Sum Expected Points

Add together the expected points from both shots to find the total expected points scored in this one-and-one situation:\[ E(\text{Total Points}) = 0.72 + 0.5184 = 1.2384 \]

Interpreting the Result

The expected number of points scored by the player in a one-and-one situation is 1.2384.

Key concepts

Basketball Strategy

In basketball, strategy is an integral part of winning games. During the endgame, when one team is trailing, they often employ a tactic known as intentional fouling. This is done to stop the clock and potentially regain possession after the opposing team shoots free throws. One key aspect of this strategy is the one-and-one free throw situation.
Here, the player must make the first shot to earn a second opportunity. If the strategy is to work effectively, it is important to understand the probabilities involved. Coaches analyze players’ free throw success rates to decide who to foul, ideally targeting players with lower success rates.
Ultimately, good basketball strategy involves a mix of player performance analysis and quick, strategic decision-making. Understanding and predicting the outcomes can improve the effectiveness of these strategic fouling techniques.

Expected Value

Expected value is a crucial concept in probability that applies to numerous situations, including sports like basketball. When calculating expected value, you're essentially finding the average expected outcome. This takes into account all possible outcomes and the probabilities of those outcomes occurring.
Here's how the concept applies to a one-and-one free throw situation:

• The player makes the first shot with a probability of 0.72, thereby securing 1 point.
• If that first shot is made, there's a 72% chance the player will make the second shot too, adding additional points.

The expected value for this scenario is computed by summing the probability of each outcome multiplied by its score. Using this information, you can predict and evaluate the average number of points a player might score over similar situations, guiding strategy decisions.
This calculated average helps coaches and players set realistic expectations and enhance game strategies.

Success Rate Calculation

Understanding success rate calculations is key when analyzing basketball shooting performance. In our example, the player has a success rate of 72% for making free throws. This is calculated by dividing the number of successful shots by the total shots taken, resulting in a decimal or percentage.
For example, if a player attempted 100 free throws and made 72, the success rate would be 72%. This probability influences predictions of success in new opportunities, such as a one-and-one free throw situation.
In situations that involve sequences or combinations of events, like getting a second shot only after making the first, compounding probabilities becomes necessary.

• Probability for first shot: 0.72
• Probability to successfully take and make the second shot: 0.72 × 0.72

By practicing and improving success rates, players can significantly impact their team's scoring opportunities. Accurate calculations ensure players and coaches use data-driven decisions while refining in-game techniques.