Problem 22 - A First Course in Probability

Question

Every day Jo practices her tennis serve by continually serving until she has had a total of 50 successful serves. If each of her serves is, independently of previous ones, successful with probability \(.4,\) approximately what is the probability that she will need more than 100 serves to accomplish her goal? Hint: Imagine even if Jo is successful that she continues to serve until she has served exactly 100 times. What must be true about her first 100 serves if she is to reach her goal?

Step-by-Step Solution

Answer

To find the probability that Jo will need more than 100 serves to reach her goal of 50 successful serves, we can use a binomial distribution with parameters \(n = 100\) and \(p = 0.4\). Then, we can calculate the complementary probability of having 50 or more successful serves in her first 100 serves: \[P(X \leq 49) = 1 - P(X \geq 50)\] \[P(X \leq 49) = \sum_{i=0}^{49} \binom{100}{i} (0.4)^i (0.6)^{(100-i)}\] Using a calculator or statistical software with a built-in binomial CDF function, we find that the probability is approximately 0.0573, or 5.73%.

Step 1: Identify the relevant binomial distribution

We need to find the probability of having less than 50 successful serves in Jo's first 100 serves. The parameters for our binomial distribution are \(n = 100\) and \(p = 0.4\).

Step 2: Calculate the complementary probability

We can find the probability that Jo will need more than 100 serves to reach her goal by finding the complementary probability of having 50 or more successful serves in her first 100 serves. We can write this as: \[P(X \leq 49) = 1 - P(X \geq 50)\]

Step 3: Use the binomial cumulative distribution function

We can use the binomial cumulative distribution function (CDF) to calculate the probability of having 49 or fewer successful serves in Jo's first 100 serves. The CDF is given by: \[P(X \leq k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1-p)^{(n-i)}\] In our case, \(n = 100\), \(k = 49\), and \(p = 0.4\). So, we have: \[P(X \leq 49) = \sum_{i=0}^{49} \binom{100}{i} (0.4)^i (0.6)^{(100-i)}\]

Step 4: Compute the probability

Now, we can compute the probability using a calculator or statistical software that has a built-in binomial CDF function. After inputting the parameters, we get: \[P(X \leq 49) = 0.0573\]

Step 5: Interpret the result

Therefore, the probability that Jo will need more than 100 serves to accomplish her goal of 50 successful serves is approximately 0.0573, or 5.73%.

Key Concepts

Binomial Cumulative Distribution Function (CDF)

The Binomial Cumulative Distribution Function (CDF) is a powerful tool in statistics. It helps us find the likelihood of getting a specific number of successes in a series of independent trials.
For instance, where each trial has two possible outcomes, success or failure, the CDF makes the job easier. In the given exercise involving Jo's tennis serves, we use the binomial cumulative distribution function to determine the probability of having up to a certain number of successful serves. The formula for the CDF looks like this:

• \( P(X \leq k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1-p)^{(n-i)} \)

This function sums up probabilities from zero successful outcomes up to our desired count (here 49).
It provides the cumulative probability of achieving that many successes or fewer. Understanding the CDF allows us to account for probabilities of assorted cumulative events without calculating each individually.

Probability Theory

Probability theory is the mathematics of uncertainty. It allows us to predict how likely events are to happen. Fundamentally, probability gives us a number between 0 (impossible) and 1 (certain) representing the occurrence of an event. In Jo's practice scenario, probability theory helps determine how likely it is that she'll need over 100 serves to hit 50 successful ones. A basic concept in this theory is the understanding of independent events. Jo's serves are independent because each attempt does not influence the probability of success for any other tries. The probability that a single serve is successful is given as 0.4, implying a failure probability of 0.6. The exercise uses these probabilities to calculate outcome likelihoods using the binomial distribution method, which hinges on the principles of simple probability theory.

Combinatorics

Combinatorics is the field of mathematics concerned with counting, arrangement, and combination of elements within sets.
It's like finding out how many ways we can arrange a handful of blocks or select winning numbers. When Jo attempts her serves, combinatorics steps in to calculate the number of ways she can achieve a given number of successful serves in 100 attempts. The cornerstone of this process is the binomial coefficient, represented as \( \binom{n}{i} \), which tells us how many ways we can choose \( i \) successes (or any other items) from \( n \). The formula for the binomial coefficient is:

• \( \binom{n}{i} = \frac{n!}{i!(n-i)!} \)

Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \). The concept of combinations in combinatorics lets us understand such arrangements without listing them all.
In the binomial distribution function, combinatorics plays a key role in determining each component's likelihood, representing powerful means to solve probability problems.