Question

A basketball player succeeds in making a basket 3 tries out of 4. How many tries are necessary in order to have probability \(>0.99\) of at least one basket?

Short answer

The player needs a minimum of 4 tries.

Step-by-step solution

Understand the Problem

A basketball player has a success rate of \( \frac{3}{4} \) or 0.75 per try. The goal is to find the number of tries (n) needed such that the probability of making at least one basket is greater than 0.99.

Probability of Failure per Try

The probability of failing to make a basket in one try is \(1 - 0.75 = 0.25\).

Probability of Failing in All n Tries

If the player fails in all \( n \) tries, the probability of failing each try independently would be \[ (0.25)^n \].

Probability of Making at Least One Basket

The probability of making at least one basket in \( n \) tries is \[ P(\text{at least one basket}) = 1 - (0.25)^n \].

Set Up Inequality

Set up the inequality to solve for \( n \): \[ 1 - (0.25)^n > 0.99 \].

Solve the Inequality

Rearrange the inequality: \[ (0.25)^n < 0.01 \]. Taking the natural logarithm of both sides, \[ \ln((0.25)^n) < \ln(0.01) \]. This simplifies to \[ n \cdot \ln(0.25) < \ln(0.01) \]. Solve for \( n \): \[ n > \frac{\ln(0.01)}{\ln(0.25)} \].

Compute the Value

Compute the value: \[ \frac{\ln(0.01)}{\ln(0.25)} = \frac{-4.6052}{-1.3863} \approx 3.32 \]. Since \( n \) must be an integer, we round up to the next whole number: \[ n = 4 \].

Key concepts

Understanding Probability

Probability measures how likely an event is to occur. It's a value between 0 and 1, where 0 means the event will not happen and 1 means it definitely will. For our basketball player, their success rate is 0.75, meaning they make a basket 75% of the time. The probability of an event not happening is 1 minus the probability of it happening. Hence, the chance of missing a basket is 1 - 0.75 = 0.25, or 25%.

Using Inequalities

Inequalities help us compare values and determine the ranges of possible solutions. For example, to ensure our basketball player makes at least one basket with a probability greater than 0.99, we used the inequality \[1 - (0.25)^n > 0.99.\] This inequality captures the requirement that we're solving for: after a certain number of tries, the player should have a more than 99% chance of making at least one basket. Inequalities are powerful in setting up such real-world problems and finding solutions.

Natural Logarithms in Problem Solving

Natural logarithms, denoted as \(\text{ln}\), are useful in many mathematical problems, particularly when dealing with exponential functions. In our problem, we used natural logarithms to solve the inequality \[(0.25)^n < 0.01\]. By taking the natural logarithm on both sides, we transformed the equation into a linear form that is easier to solve: \[n \times \text{ln}(0.25) < \text{ln}(0.01).\] This gave us a more manageable way to isolate and find the value of \(n\).

Application to Basketball Statistics

Basketball statistics provide valuable insights into players' performances. For instance, knowing that our player has a 75% success rate is crucial for making predictions about future attempts. By applying probability, inequalities, and natural logarithms, we were able to determine that they need at least 4 tries to have a higher than 99% chance of scoring at least once. This kind of analysis can be extended to other areas in sports and real-life scenarios, highlighting the importance and utility of mathematical concepts in everyday decision-making.