1. Identify the problem type: This is a binomial probability problem where we want to find the probability of exactly 2 successes (made free throws) out of 6 attempts.
2. Define the variables:
   • \(n = 6\) (number of trials)
   • \(k = 2\) (number of successes)
   • \(p = 0.60\) (probability of success on each trial)
3. Use the binomial probability formula:
   \[P( X = k) = \binom { n} { k} p^ k ( 1- p) ^ { n- k} \]
   where \(\binom { n} { k} \) is the binomial coefficient.
4. Calculate the binomial coefficient:
   \[\binom { 6} { 2} = \frac { 6! } { 2! ( 6- 2) ! } = \frac { 6 \times 5} { 2 \times 1} = 15\]
5. Calculate \(p^ { k} \) and \(( 1- p) ^ { ( n- k) } \):
   \[p^ k = 0.60^ 2 = 0.36\]
   \[( 1- p) ^ { n- k} = 0.40^ 4 = 0.0256\]
6. Combine the values:
   \[P( X = 2) = 15 \times 0.36 \times 0.0256 = 0.13824\]
7. Convert to a percentage:
   \[0.13824 \approx 13.8\% \]
   Thus, the probability that the player makes exactly two out of six free throws is 13.8%.

Supplemental Knowledge

The problem you're dealing with involves calculating the probability of a specific number of successes in a fixed number of independent trials, where each trial has the same probability of success. This is a classic example of a binomial distribution.
In a binomial distribution, the probability \(P( X = k) \) of getting exactly \(k\) successes in \(n\) trials is given by:
\[P( X = k) = \binom { n} { k} p^ k ( 1- p) ^ { n- k} \]
where:

• \(n\) is the total number of trials,
• \(k\) is the number of successful trials,
• \(p\) is the probability of success on an individual trial,
• \(\binom { n} { k} \) (read as "n choose k") is the binomial coefficient, calculated as \(n! / ( k! ( n- k) ! ) \).
  For instance, if you were to flip a fair coin 10 times and wanted to know the probability of getting exactly 4 heads, you would use this formula with \(n = 10\), \(k = 4\), and \(p = 0.5\).

Real-World Applications

Consider you're participating in a carnival game where the object of throwing rings onto bottles has an uncertain chance of landing on one; using binomial distribution you can predict your chances of success from past experience and determine your chance at winning prizes based on how many throws are allowed based on past performance. Not only is this concept applicable in games but also manufacturing companies needing to predict defect rates across batches of product produced.

Probability and statistics are powerful tools that help us make sense of uncertainty in various aspects of life, from sports performance to quality control in manufacturing. If you're eager to master these concepts or have other mathematical queries, UpStudy offers specialized calculators like our Statistics and Probability Calculator that can assist you with complex calculations and deepen your understanding. Explore UpStudy today and elevate your learning experience with our comprehensive resources!