Problem 25 - Applied Probability

Question

Mike and Harry have a basketball shooting contest. \- Mike shoots 10 ordinary free throws, worth two points each, with probability 0.75 of success on each shot. \- Harry shoots 12 "three point" shots, with probability 0.40 of success on each shot. Let \(X, Y\) be the number of points scored by Mike and Harry, respectively. Determine \(P(X \geq 15)\), and \(P(Y \geq 15), P(X \geq Y)\)

Step-by-Step Solution

Answer

To summarize, calculate the cumulative probabilities for Mike and Harry scoring at least 15 points using their respective binomial distributions. For comparing scores, calculate the joint probability for all scenarios where Mike's score exceeds Harry's.

Step 1: Calculate Probability for Mike

Since Mike shoots 10 times and each successful shot is worth 2 points, the number of points Mike scores follows a binomial distribution with parameters n=10 (trials) and p=0.75 (success probability). We want to find the probability that Mike scores at least 15 points, which means he needs at least 8 successful shots (as 8 shots * 2 points = 16 points). Calculate the cumulative probability for 8, 9, and 10 successful shots and sum them up.

Step 2: Calculate Probability for Harry

Similar to Mike, Harry's score follows a binomial distribution, but with n=12 and p=0.40. Harry needs at least 5 successful shots to score at least 15 points (5 shots * 3 points = 15 points). Calculate the cumulative probability for 5, 6, 7, ..., 12 successful shots and sum them up.

Step 3: Determine Probability of Mike Scoring More than Harry

For this, we need to calculate the probability distribution of Mike's and Harry's scores, then sum the probabilities where Mike's score exceeds Harry's. This involves summing over all combinations of Mike's and Harry's scores where Mike's score is higher. Due to the complexity, this might be more neatly done with software or by using a more advanced combinatorial approach.

Key Concepts

Cumulative Probability

When we talk about cumulative probability, we are referring to the likelihood that a random variable falls within a specified range or below a certain value. In the context of our basketball shooting contest, for the players to find success, they aim to score a minimum number of points. To calculate the cumulative probability for Mike to score at least 15 points, we sum up the probabilities of him scoring 16, 18, or the maximum 20 points. Similarly, for Harry, we consider the probabilities of scoring 15 points and beyond up to the maximum of 36 points.

In technical terms, if we have a discrete random variable like the number of successful shots, the cumulative probability up to a certain point is the sum of probabilities of all outcomes up to and including that point. Mathematically, for a certain number of points say 'k', the cumulative probability is denoted as \( P(X \leq k) \), where 'X' is the random variable representing the score. For our case, it's the cumulative distribution function (CDF) that comes into play, summing the probabilities for the necessary number of successful shots to reach a target. This concept is crucial because it helps us to understand not just the likelihood of a single outcome, but the collective likelihood of multiple outcomes.

Probability Distribution

Moving on to the probability distribution, this is a mathematical function that provides the probabilities of occurrence of different possible outcomes for an experiment. It's like a map that assigns a probability to each possible event. In a binomial setting, the distribution will tell us the probabilities of achieving 'r' successes in 'n' trials, given the success probability 'p' in each trial.

For example, Mike's performance can be represented by a binomial distribution with a success probability of 0.75 in 10 trials. This distribution will show us the likelihood of Mike making anywhere from 0 to 10 successful shots. The function we use to represent this probability distribution is the probability mass function (PMF), which in the binomial case, is given by the formula:\[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \] where \( \binom{n}{r} \) is the binomial coefficient, indicating the number of ways to choose 'r' successes out of 'n' trials. Understanding the probability distribution is essential because it informs us about the spread and tendencies of our random variable, in this case, the players' scores.

Binomial Theorem

Lastly, the binomial theorem is a fundamental principle that can help us solve problems related to binomial distributions. It tells us how to expand expressions that raise a binomial to a power. The theorem expresses the expansion as:\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] where \( a \) and \( b \) are any numbers, and \( n \) is a positive integer. Each term in the expansion has a coefficient known as a binomial coefficient, represented by \( \binom{n}{k} \), which corresponds to the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

In probability, the binomial theorem provides the foundation for the binomial distribution's probability mass function, as mentioned previously. The coefficients represent the number of ways to achieve a specific number of successes (like successful shots in the game). This links directly to the concept of probability distribution, reinforcing the idea that mathematics often involves working with interconnected concepts to arrive at a solution to a problem. Grasping the binomial theorem not only enhances one's algebraic skills but also deepens understanding of probability and combinatorics, both of which are essential to solving binomial distribution problems like those posed by the basketball shooting contest of Mike and Harry.