Question

The paper "Action Bias among Elite Soccer Goalkeepers: The Case of Penalty Kicks" ( Journal of Economic Psychology [2007]: \(606-621\) ) presents an interesting analysis of 286 penalty kicks in televised championship soccer games from around the world. In a penalty kick, the only players involved are the kicker and the goalkeeper from the opposing team. The kicker tries to kick a ball into the goal from a point located 11 meters away. The goalkeeper tries to block the ball from entering the goal. For each penalty kick analyzed, the researchers recorded the direction that the goalkeeper moved (jumped to the left, stayed in the center, or jumped to the right) and whether or not the penalty kick was successfully blocked. Consider the following events: \(L=\) the event that the goalkeeper jumps to the left \(C=\) the event that the goalkeeper stays in the center \(R=\) the event that the goalkeeper jumps to the right \(B=\) the event that the penalty kick is blocked Based on their analysis of the penalty kicks, the authors of the paper gave the following probability estimates: $$ \begin{array}{rrr} P(L)=0.493 & P(C)=0.063 & P(R)=0.444 \\ P(B \mid L)=0.142 & P(B \mid C)=0.333 & P(B \mid R)=0.126 \end{array} $$ a. For each of the given probabilities, write a sentence giving an interpretation of the probability in the context of this problem. b. Use the given probabilities to construct a "hypothetical 1000" table with columns corresponding to whether or not a penalty kick was blocked and rows corresponding to whether the goalkeeper jumped left, stayed in the center, or jumped right. (Hint: See Example 5.14) c. Use the table to calculate the probability that a penalty kick is blocked. d. Based on the given probabilities and the probability calculated in Part (c), what would you recommend to a goalkeeper as the best strategy when trying to defend against a penalty kick? How does this compare to what goalkeepers actually do when defending against a penalty kick?

Short answer

Interpreting probabilities suggests goalkeepers tend to jump left or right most often. A hypothetical 1000 table translates these percentages into an estimated number of occurrences in 1000 penalty kicks. The calculated probability for a penalty kick being blocked is approximately \(0.162\) or 16.2%. The best strategy for goalkeepers, contrary to their habit of jumping left or right, may be to stay in the center more often, as it has a higher block rate of 33.3%.

Step-by-step solution

Interpret the probabilities

Probabilities given are: \(P(L)=0.493\) indicates that goalkeepers jump to the left 49.3% of the time. \(P(C)=0.063\) shows that goalkeepers stay in the center 6.3% of the time. \(P(R)=0.444\) implies that goalkeepers jump to the right 44.4% of the time. Conditional probabilities: \(P(B \mid L)=0.142\) shows that when a goalkeeper jumps left, the penalty kick is blocked 14.2% of the time. \(P(B \mid C)=0.333\) implies that when a goalkeeper stays in the center, penalty kick is blocked 33.3% of the time. \(P(B \mid R)=0.126\) suggests that when a goalkeeper jumps right, the penalty kick is blocked 12.6% of the time.

Construct a hypothetical 1000 table

Given probabilities can be used to construct a hypothetical table for 1000 penalty kicks. See how many times it would be expected for each event to occur out of 1000 times. For example, out of 1000 penalty kicks, the goalkeeper would be expected to jump left about \(0.493 * 1000 = 493\) times, they'd stay in the center for \(0.063 * 1000 = 63\) times and so on.

Calculate the probability

Use the table to calculate the probability that a penalty kick is blocked. To do this, add up the 'blocked' entries in your table and divide by 1000, which will be the sum of \(493 * 0.142 + 63 * 0.333 + 444 * 0.126)\) divided by 1000.

Give a recommendation

To recommend a strategy to a goalkeeper, look at which action had the highest blocking percentages and compare it to what goalkeepers actually do. Note that while goalkeepers seem to jump left or right more often, staying in the center actually has a higher block rate, so that could be a recommended strategy.

Key concepts

Probability Estimates

The concept of probability estimates plays a vital role in understanding events and their likely outcomes. It is essential in various fields such as sports, where coaches and players make strategic decisions based on the possible chances of certain plays. For instance, in the context of penalty kicks in soccer, researchers provided specific probabilities for each direction the goalkeeper might dive and the associated success rates.

Consider the given data: the probability of a goalkeeper jumping to the left (\(P(L)\)) is 49.3%, to the center (\(P(C)\)) is 6.3%, and to the right (\(P(R)\)) is 44.4%. These probabilities are the basis for estimating how likely each event is to occur. More importantly, the conditional probabilities such as \(P(B \mid L)\) which is 14.2%, suggests that if the goalkeeper chooses to jump left, the likelihood of blocking the kick is that specific percentage. Understanding these estimates can significantly influence a goalkeeper's decision-making process, leading to a more strategic approach to defending penalty kicks.

Economic Psychology

Economic psychology examines how psychological factors influence economic behaviors, such as the decision-making process in sports. The concept is well illustrated by the action bias phenomenon among soccer goalkeepers facing penalty kicks. Despite the statistics indicating a higher success rate of staying in the center, goalkeepers often jump left or right. This bias is rooted in psychological pressures and the expectation to physically react, often leading to suboptimal decisions.

In other words, players and coaches are influenced not only by the statistical probabilities but also by the weight of expectation and the perceived need for action. This area of study can provide insights into why goalkeepers might choose strategies that deviate from what probability estimates would suggest is best. Addressing these psychological pressures and educating players about the actual statistics can encourage better decision-making under pressure.

Sports Statistics

Sports statistics involve the collection and analysis of data related to sports performance. The statistical information provides a foundation for evaluating player actions, forming strategies, and predicting future outcomes. In our soccer penalty kick scenario, the statistical analysis draws a clear picture of the tendencies of goalkeepers and the success rates associated with their decisions.

By constructing a hypothetical table based on 1000 penalty kicks, we illustrate how often goalkeepers choose a particular direction and how successful they are. Through such sports statistics, it is evident that although the center position is chosen less frequently, it shows a significantly higher success rate of blocks (33.3%). This analysis can induce a change in coaching tactics and player behavior, all with the goal of optimizing performance and outcomes on the field.