Problem 75 - A First Course in Probability

Question

A fair coin is continually flipped until heads appears for the 10 th time. Let \(X\) denote the number of tails that occur. Compute the probability mass function of \(X\)

Step-by-Step Solution

Answer

The probability mass function for the number of tails that appear before the 10th head in a series of coin flips using a fair coin is given by: \(P(X = x) = \binom{x+9}{x}\left(\frac{1}{2}\right)^{x+10}\)

Step 1: Understand the Negative Binomial Distribution

The negative binomial distribution is a discrete probability distribution for the number of trials required for a fixed number of successes. In our case, we need 10 heads, and we want to find the number of tails occurring before the 10th head. The probability mass function of a negative binomial distribution is given by: \(P(X = x) = \binom{x+r-1}{x}p^r(1-p)^x\) Where - \(X\) represents the number of failures (tails) before the \(r\)th success, - \(r\) is the number of successes (10 heads), - \(p\) is the probability of success (probability of flipping a head, which is 1/2), and - \(x\) is the number of trials.

Step 2: Set Parameters

In our problem, the parameters for the negative binomial distribution are: - \(r = 10\) (number of successes), - \(p = 1/2\) (probability of flipping a head). We want to find the probability of having \(x\) tails before the 10th head.

Step 3: Calculate the Probability Mass Function

Using the parameters from Step 2 and the negative binomial distribution formula, we can now calculate the probability mass function: \(P(X = x) = \binom{x+10-1}{x}\left(\frac{1}{2}\right)^{10}\left(\frac{1}{2}\right)^{x}\) Simplifying the formula, we get: \(P(X = x) = \binom{x+9}{x}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{2}\right)^{10}\) Now we need to express this as a probability mass function for the given problem: \(P(X = x) = \binom{x+9}{x}\left(\frac{1}{2}\right)^{x+10}\) This is the probability mass function of \(X\), the number of tails that occur before the 10th head in a series of coin flips using a fair coin.

Key Concepts

Probability Mass Function

The probability mass function (PMF) is a fundamental concept in the study of discrete probability distributions. It defines the probability of our discrete random variable, which in layman's terms means it tells us how likely each possible outcome is for a scenario that has a finite number of possible results.

The PMF is denoted by the function:
\[ P(X = x) \]
where \( P \) is the probability and \( X \) is the random variable taking on the value \( x \). In the context of a coin flip, where the coin is flipped repeatedly until a certain number of heads are obtained, the PMF allows us to calculate the probability of getting a specific number of tails before achieving the desired number of heads. It's crucial for students to recognize that PMFs are only used for discrete scenarios – they can count the outcomes, like the number of tails in our coin flip example.

While calculating the PMF, any real-world understanding of the context can greatly improve the grasp of the concept. In our exercise, as the coin flips are independent events, the probability of getting a tail in each flip remains constant, which simplifies the application of the PMF.

Discrete Probability Distribution

When discussing discrete probability distributions, we're looking at a system where outcomes are counted and not measured. Think of flipping coins, rolling dice, or drawing cards. Because you can list all the possible outcomes, the distribution of probabilities across these outcomes can be described using the PMF mentioned previously.

In the given exercise with coin flips, the possible outcomes are discrete because we can't have half a flip or 1.35 tails. We either have a tail, or we don't, signifying a success or failure. Each outcome has an associated probability, and the sum of all these probabilities equals one – a core principle in probability theory known as the normalization condition.

Understanding this concept is key when dealing with exercises in probability. It forms the basis for more complex topics and helps build foundational knowledge about how and why certain outcomes occur and their likelihood.

Binomial Coefficient

The binomial coefficient is symbolized by
\[ \binom{n}{k} \]
and represents the number of ways to pick \( k \) successes out of \( n \) trials, without worrying about the order of the successes. This calculation is an essential part of many probability formulas, including the negative binomial distribution from our exercise.

The binomial coefficient can be computed using factorials as follows:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
where \(!\) denotes the factorial operation, which means multiplying a series of descending natural numbers.

In our coin flip scenario, where we count the number of tails (failures) before the 10th head (success), we use binomial coefficients to account for all possible sequences of tails and heads that could occur. It's a measure that gives significant depth to understanding probabilities in various combinations and is a building stone for grasping more complex statistical concepts.