Question

Two stars of the LA Lakers are very different when it comes to making free throws. ESPN.com reports that Kobe Bryant makes \(85 \%\) of his free throw shots while Lamar Odum makes \(62 \%\) of his free throws. \(^{5}\) Assume that the free throws are independent and that each player shoots two free throws during a team practice. a. What is the probability that Kobe makes both of his free throws? b. What is the probability that Lamar makes exactly one of his two free throws? c. What is the probability that Kobe makes both of his free throws and Lamar makes neither of his?

Short answer

Question: Calculate the probabilities for the following events during a basketball team practice where each player, Kobe Bryant and Lamar Odum, shoot two free throws each. (a) Kobe makes both of his free throws. (b) Lamar makes exactly one of his two free throws. (c) Kobe makes both of his free throws while Lamar misses both of his. Given that Kobe has an 85% free throw shooting percentage, and Lamar has a 62% free throw shooting percentage. Answer: (a) The probability that Kobe makes both of his free throws is approximately 0.7225. (b) The probability that Lamar makes exactly one of his two free throws is approximately 0.4788. (c) The probability that Kobe makes both of his free throws and Lamar makes neither of his is approximately 0.3455.

Step-by-step solution

Calculate the probability that Kobe makes both of his free throws

We are given that Kobe makes 85% of his free throws. Since the free throws are independent events, the probability of him making both free throws is the product of the probabilities of him making each free throw. So, the probability that Kobe makes both of his free throws is: 0.85 * 0.85 ≈ 0.7225

Calculate the probability that Lamar makes exactly one of his two free throws

We are given that Lamar makes 62% of his free throws. Since the free throws are independent events, there are two ways Lamar can make exactly one free throw: make the first one and miss the second one or miss the first one and make the second one. We need to calculate the probability of each scenario and sum them up. So, the probability that Lamar makes exactly one of his two free throws is: (0.62 * 0.38) + (0.38 * 0.62) ≈ 0.4788

Calculate the probability that Kobe makes both of his free throws and Lamar makes neither of his

To find the probability that Kobe makes both of his free throws and Lamar misses both of his, we can multiply the probabilities that we calculated in steps 1 and 2. So, the probability that Kobe makes both of his free throws and Lamar makes neither of his is: 0.7225 * 0.4788 ≈ 0.3455 Therefore, the solutions to the exercise are: a. The probability that Kobe makes both of his free throws is approximately 0.7225. b. The probability that Lamar makes exactly one of his two free throws is approximately 0.4788. c. The probability that Kobe makes both of his free throws and Lamar makes neither of his is approximately 0.3455.

Key concepts

Independent Events

In the realm of probability, the concept of independent events is crucial. Simply put, two events are independent when the occurrence of one event does not affect the probability of the other occurring. In basketball free throws, each shot is considered an independent event. The result of one shot does not influence the outcome of the next one. This means, for example, if a player is excellent at making 85% of his free throws, the outcome of the next shot still has the same 85% probability as the previous shot.

• Example: If Kobe Bryant makes his first shot, it doesn't change the likelihood of making the second. Each has the same statistical chance.
• Mathematically, if event A and event B are independent, the probability of both events occurring is the product of their individual probabilities, i.e., \( P(A \text{ and } B) = P(A) \times P(B) \).

Understanding this concept is vital when calculating the likelihood of players making consecutive shots, which is a common task in basketball statistics.

Free Throw

A free throw in basketball is a type of shot made from a designated line on the court with no defense against the shooter. Successful free throws can significantly impact the outcome of the game by adding to the team's score.
They provide a unique scenario for probability analysis because each free throw is an isolated event, usually considered independent.

• The player's free throw percentage is used to determine their likelihood of making a shot.
• Kobe Bryant, known for his shooting skills, typically makes 85% of his free throws. This statistic indicates his free throw probability.
• On the other hand, Lamar Odum's 62% suggests a lower probability of scoring from the free throw line, highlighting differences in player performance.

These statistics allow us to apply probability calculations to predict outcomes during practice or actual games.

Probability Calculation

Calculating probability involves determining how likely an event is to occur. In the case of basketball free throws, we look at individual player's shooting percentages.
Here's a breakdown of common scenarios:

• Kobe Making Both Free Throws: Since each shot is independent, the probability that Kobe makes both shots is found by multiplying his success rate for each shot, \(0.85 \times 0.85 = 0.7225\).
• Lamar Making Exactly One Free Throw: There are two possible sequences: making the first shot and missing the second, or missing the first and making the second. Both scenarios have a probability of \(0.62 \times 0.38\), and we sum these probabilities to get \((0.62 \times 0.38) + (0.38 \times 0.62) = 0.4788\).
• Kobe Making Both, Lamar Missing Both: To find this combined probability, multiply Kobe's success rate by the probability of Lamar missing both shots \(0.7225\) and \((0.38 \times 0.38 = 0.1444)\), resulting in \( 0.7225 \times 0.1444 = 0.1043\).

These calculations help in predicting player performance outcomes and are key to evaluating potential game strategies.

Basketball Statistics

Basketball statistics provide valuable insights into players' performance and the team's overall effectiveness. They help in forming strategies and making informed decisions during games. Some common statistical areas include:

• Shooting Percentage: This metric evaluates how efficiently a player scores from the field, free throw line, or three-point range. Kobe Bryant's 85% free throw statistic shows high scoring efficiency.
• Free Throw Percentage: This is the ratio of successful free throws a player makes out of total attempts, important for predicting potential points from fouls.
• Analyzing Patterns: By examining data over multiple games or practices, teams can identify strengths and weaknesses, adjusting training or strategies accordingly.
• Independent Events Consideration: Recognizing that each shot is independent means stats can pinpoint trends rather than random occurrences.

Understanding and applying basketball statistics aids in achieving competitive advantages and fostering player improvement through targeted practice.