Problem 114 - Algebra and Trigonometry Real Mathematics, Real People

Question

Consider \(n\) independent trials of an experiment in which each trial has two possible outcomes, success or failure. The probability of a success on each trial is \(p\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment. The probability of a baseball player getting a hit during any given time at bat is \(\frac{1}{4} .\) To find the probability that the player gets three hits during the next 10 times at bat, evaluate the term $$_{10} C_{3}\left(\frac{1}{4}\right)^{3}\left(\frac{3}{4}\right)^{7}$$ in the expansion of \(\left(\frac{1}{4}+\frac{3}{4}\right)^{10}\).

Step-by-Step Solution

Answer

The probability that the player gets three hits during the next 10 times at bat is approximately 0.25028.

Step 1: Understand the binomial expression

The formula for calculating a binomial probability is given as \(_nC_k p^k q^{n-k}\), where \(_nC_k\) is the binomial coefficient, \(p\) is the probability of success, \(q\) is the probability of failure, \(k\) is the number of successes and \(n\) is the number of trials. This expression is also a term in the expansion of \((p+q)^n\).

Step 2: Substitute the given values into the expression

We are given \(n = 10\) , \(k = 3\) , \(p =\frac{1}{4}\), and \(q = 1 - p =\frac{3}{4}\). Substituting these values into the binomial expression will give us: \(_{10}C_3 \left(\frac{1}{4}\right)^3\left(\frac{3}{4}\right)^7\)

Step 3: Compute the binomial coefficient

The binomial coefficient \(_nC_k\) is calculated as \(\frac{n!}{k!(n - k)!}\), where \(n!\) means 'n factorial', which is the product of all positive integers up to \(n\). Calculating this for the given values, we get \(_{10}C_3 = \frac{10!}{3!(10 - 3)!} = 120\)

Step 4: Evaluate the entire expression

Substitute the computed binomial coefficient into the expression to get \(120 \left(\frac{1}{4}\right)^3\left(\frac{3}{4}\right)^7\). Evaluating this expression should yield the final probability.

Key Concepts

Binomial Coefficient

Imagine you're at a buffet, and you can only choose a certain number of dishes from a larger selection. The binomial coefficient represents how many different ways you can make those choices. Mathematically, the binomial coefficient, denoted as C_k, details the number of combinations of choosing k items from a larger set of n items, without caring about the order.

In probability, this tells us how many different ways an event can occur. It's a pivotal part of the binomial probability formula and is calculated using factorials. So, for our baseball player wishing to get three hits out of ten at-bats, we calculate the binomial coefficient as C_k = \(\frac{n!}{k!(n - k)!}\) which helps us predict his chances with precision.

Factorial

A factorial, symbolized by an exclamation point (!), is a product of all positive integers up to a certain number. So, when you see n!, think of it as a domino effect of multiplication: \(n \times (n-1) \times (n-2) \times ... \times 1\). It's the multiplication version of a countdown. Factorials are not just mathematical fun; they're fundamental when determining the binomial coefficient, which we've learned is essential in computing probabilities in a binomial experiment.

In our example, 10! is critical for finding out how many ways our player can hit three times in ten at-bats. Understanding factorials means grasping the scale of possibilities in combinatorial scenarios.

Binomial Theorem

The binomial theorem is like a blueprint for expanding expressions raised to a power, such as \((a + b)^n\). It tells us that we can express this as a sum of terms involving binomial coefficients. Each term represents a possible scenario in our binomial experiment.

It's not just an abstract notion—this theorem is how we can quickly find any term in the expanded form without multiplying everything out the long way. For instance, the term we seek to evaluate for the baseball player's probability is a direct application of the binomial theorem to the powers of \(\frac{1}{4}\) and \(\frac{3}{4}\) in \(\bigg(\frac{1}{4}+\frac{3}{4}\bigg)^{10}\).

Binomial Experiment

Consider a binomial experiment as a series of 'yes or no' questions. In each trial, there are only two possible outcomes, like flipping a coin, where you can only get heads or tails. For our baseball player, each at-bat is a trial with two outcomes: a hit or no hit.

A bona fide binomial experiment has a fixed number of trials, each trial has the same probability of success, the trials are independent, and we're interested in the number of successes. Knowing that our experiment is binominal allows us to use specific tools, like the binomial theorem, to compute the likelihood of different outcomes.

Probability of Success

In a binomial experiment, the probability of success is the cherry on top. It's denoted by p and represents the chance that each trial results in a 'yes' answer to your question. In the case of the baseball player, the probability of success is the chance of getting a hit, which is \(\frac{1}{4}\) or 25%. The counterpart, the probability of failure, is equally important, marked as q and subtracted from 1.

Picturing these probabilities can help understand why some events are more likely than others. It also clarifies why the probability of getting three hits isn't just three times the probability of getting one hit; it's a combination of several probabilities and combinations working together.