Problem 4 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

You toss \(n\) coins, each showing heads with probability \(p\), independently of the other tosses. Each coin that shows tails is tossed again. Let \(X\) be the total number of heads. a. What type of distribution does \(X\) have? Specify its parameter(s). b. What is the probability mass function of the total number of heads \(X ?\)

Step-by-Step Solution

Answer

X is Binomially distributed with parameters \( n \) and \( 1-(1-p)^2 \). The PMF is \[ f(x) = \binom{n}{x} (1-(1-p)^2)^x (1-p)^{2(n-x)} \].

Step 1: Analyze the Problem

We need to determine the type of probability distribution that fits the scenario where we toss each coin twice potentially. A first toss gives a head with probability \( p \), and if it is a tail (with probability \( 1-p \)), the coin is tossed again for a second chance at a head.

Step 2: Identify Distribution Type

Since each coin toss can be modeled as a Bernoulli trial, and a coin that initially lands tails is retossed, the number of heads in the second trial is also a Bernoulli trial. This setup models a Geometric distribution, where each coin's successive tosses form a geometric sequence until a head appears. The overall result for total heads in a pool of \( n \) coins that each resolve separately is a Binomial distribution.

Step 3: Define Parameters for Distribution

In our scenario, the distribution is a Binomial distribution because we have \( n \) independent Bernoulli trials (each coin), albeit with repeated attempts for tails. The parameter is the probability of ultimately getting a head, which is calculated as \( 1-(1-p)^2 \), where the second term is the chance of two consecutive tails and no head.

Step 4: Write the Probability Mass Function

The probability mass function (PMF) of a Binomial distribution \( X \) defined as \( \text{Binomial}(n, 1-(1-p)^2) \) is given by:\[ f(x) = \binom{n}{x} \cdot \left(1-(1-p)^2\right)^x \cdot \left((1-p)^2\right)^{n-x}\] where \( x \) is the number of heads (successes) observed.

Key Concepts

Binomial Distribution

A Binomial Distribution is commonly used to model the number of successes in a fixed number of independent Bernoulli trials. These trials are experiments that have two possible outcomes: success or failure. In our exercise, each coin toss is a Bernoulli trial with a chance of landing heads (success) or tails (failure). The main features of a Binomial Distribution include:

• Fixed Number of Trials: Here, the trials are fixed by the number of coins, meaning there's a determined number of attempts to achieve a head on any coin.
• Two Possible Outcomes: Each coin can show heads or tails, making it a success or failure.
• ID Properties: Each outcome of the coin toss does not depend on previous outcomes.
• Probability of Success: This is defined as the likelihood of getting a head, corrected for two possible tosses due to the retoss.

Overall, the scenario in our exercise fits a Binomial Distribution model because we handle multiple independent coins and count the number of heads across these tosses.

Probability Mass Function

The Probability Mass Function (PMF) is a fundamental tool in understanding the behavior of discrete random variables, like the number of heads in our coin-tossing scenario. The PMF provides the probability that a discrete random variable is exactly equal to some value. For a Binomial Distribution, this is expressed mathematically as:
\[ f(x) = \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}\]
- Here, \(x\) is the number of heads (successes) observed.
- \(\binom{n}{x}\) represents the number of ways to choose \(x\) successes in \(n\) trials.
- \(p\) is the probability of observing a head (success).
- \((1-p)\) is the probability of a tail (failure), adjusted in our exercise for potential retossing. This is calculated as the probability of two consecutive tails.

The PMF allows us to calculate the probability of any given number of heads appearing across all coin tosses, harnessing the reshaped probability value after reattempts.

Bernoulli Trial

A Bernoulli Trial is the simplest form of a probabilistic experiment and is characterized by having only two outcomes: success or failure. In the context of our exercise, each coin toss represents a Bernoulli Trial. A coin shows heads with probability \(p\) and tails with probability \(1-p\). Let's understand key aspects of Bernoulli Trials:

• Binary Outcome: Each trial results in one of two outcomes, similar to a win or lose, yes or no situation.
• Independent Trials: The likelihood of each outcome does not change or get influenced by other trials.
• Consistency: The probability of success \(p\) remains constant throughout the trials.

The concept of Bernoulli Trials is essential for forming Binomial and Geometric Distributions when dealing with series of repeated binary experiments like coin tosses.

Geometric Distribution

When exploring probability problems, a Geometric Distribution becomes important when determining the number of trials needed for the first success. It emerges distinctly in our exercise through retossing coins that initially land tails until we eventually see a head. Here are some core ideas behind Geometric Distribution:

• First Success Focus: It accounts for scenarios where we're interested in the number of attempts before achieving the first success.
• Memoryless Property: The probability of success on the next trial is always the same, regardless of prior failures.
• Independent Trials: Similar to Bernoulli Trials, each attempt is independent of others.

In the context of our problem, although each coin is resolved in a Binomial manner, the process of retossing for the first head reflects an aspect of a Geometric Distribution, echoing concepts from the geometric model through repeated trials for a first success.