Question

DRAWING CONCLUSIONS The Redbirds trail the Bluebirds by one goal with 1 minute left in the hockey game. The Redbirds' coach must decide whether to remove the goalie and add a frontline player. The probabilities of each team scoring are shown in the table. \begin{tabular}{|l|c|c|} \cline { 2 - 3 } \multicolumn{1}{c|}{} & Goalie & No goalie \\ \hline Redbirds score & \(0.1\) & \(0.3\) \\ \hline Bluebirds score & \(0.1\) & \(0.6\) \\ \hline \end{tabular} a. Find the probability that the Redbirds score and the Bluebirds do not score when the coach leaves the goalie in. b. Find the probability that the Redbirds score and the Bluebirds do not score when the coach takes the goalie out. c. Based on parts (a) and (b), what should the coach do?

Short answer

a) The probability that the Redbirds score and the Bluebirds do not score when the goalie is in the game is \(0.09\). b) The probability that the Redbirds score and the Bluebirds do not score when the goalie is out of the game is \(0.12\). c) The coach should remove the goalie.

Step-by-step solution

Calculate probability with the goalie in the game

The probability of the Redbirds scoring and the Bluebirds not scoring when the goalie is in the game is given by the multiplication of the individual probabilities. For the Redbirds to score, the probability is \(0.1\), and for the Bluebirds not to score, the probability is \(1 - 0.1 = 0.9\). So, the required probability is \((0.1) * (0.9) = 0.09\).

Calculate probability with the goalie out of the game

The probability of the Redbirds scoring and the Bluebirds not scoring when the goalie is out of the game is similar to the above step. For the Redbirds to score, the probability is \(0.3\), and for the Bluebirds not to score, the probability is \(1 - 0.6 = 0.4\). As a result, the required probability is \((0.3) * (0.4) = 0.12\).

Choose the best option for the coach

Based on the previous calculations, the probability of the Redbirds scoring and the Bluebirds not scoring is seen to be higher when the goalie is out of the game (\(0.12\) vs. \(0.09\)). Hence, it would be a better option for the Redbirds' coach to remove the goalie and add a frontline player.

Key concepts

Conditional Probability

Understanding conditional probability is essential when calculating the chances of an event occurring, given that another event has already taken place. In sports, coaches often assess probabilities to make strategic decisions during a game, such as in our example where a hockey coach contemplates removing the goalie to score.

When we calculated the likelihood of the Redbirds scoring without the goalie and the Bluebirds not scoring, we were actually applying conditional probability. The probability of one team scoring is conditional upon the other team's goalie's presence. In mathematical terms, conditional probability is often represented as P(A|B), meaning the probability of event A happening given that B has occurred.

To improve comprehension of this concept, it's insightful to realize that conditional probabilities can dramatically change depending on the circumstances of the game (such as the time left or the current score), which makes calculating these scenarios vital for in-game decisions.

Multiplication Rule of Probability

The multiplication rule of probability allows us to find the probability that two events will happen together. This becomes exceptionally important in sports where multiple outcomes can occur simultaneously, and strategies are often based on these probabilities.

As seen in our hockey game example, we calculated the probability of the Redbirds scoring and the Bluebirds not scoring. This calculation was done by multiplying the individual probabilities of each event. This is an application of the multiplication rule, which generally states that if events A and B are independent, then the probability of both events occurring is P(A and B) = P(A) × P(B).

For a clearer understanding, consider an illustrative example: If a basketball player has a 50% chance of making a free throw and they take two free throws, the multiplication rule can be used to find the probability that they make both shots - (0.5) * (0.5) = 0.25, or a 25% chance.

Probability Decision Making

When it comes to decision making, probability plays a crucial role in determining the best course of action. In sports, coaches use these calculations to make choices under pressure that can ultimately influence the outcome of the game.

In our hockey scenario, the coach used probability to decide whether to keep the goalie in or take him out. By calculating the chances of scoring and preventing the opponent from scoring under both conditions, the coach was able to make an informed decision. This evident in the solution where the Redbirds had a higher probability of scoring when the goalie was removed (12% vs. 9%).

Probability decision making is not just about the numbers; it's also about weighing the risks and rewards. Coaches and players often have to consider not only the statistical probabilities but also the nuances of the game, such as momentum, player fatigue, and psychological factors, which might not be easily quantifiable.