Problem 41 - An Introduction to Mathematical Statistics and Its Applications

Question

A fair coin is tossed three times. Let \(X\) be the number of heads in the tosses minus the number of tails. Find \(p_{X}(k)\).

Step-by-Step Solution

Answer

Probability mass function \(p_{X}(k)\) of \(X\) will be calculated by counting number of events where number of heads minus number of tails equals to \(k\), each such event happening with probability \(0.5^3\). There will be different \(p_{X}(k)\) values for each \(k\) in [-3, 3].

Step 1: Identify possible values for \(X\)

First, identify all possible outcomes of three coin flips and calculate the value of \(X\) (the number of heads minus the number of tails) for each. There can be 0, 1, 2 or 3 heads and similarly 0, 1, 2 or 3 tails.

Step 2: Identify probability for each possible value of \(X\)

Now, for a given \(X=i\), calculate the number of events using binomial coefficient where number of heads minus number of tails equals to \(i\). Multiply this with \(0.5^3\) which is the probability for one event since the coin is fair and each event consists of 3 independent flips.

Step 3: Calculate \(p_{X}(k)\) for each possible value of \(X\)

Perform the calculation mentioned in step 2 for each possible \(X\) which can range from -3 to 3 here. \(p_{X}(k)\) is obtained as the result. This is the probability of \(X=i\) for each \(i\) in [-3, 3].

Key Concepts

Binomial Coefficient

Understanding the binomial coefficient is fundamental when dealing with exercises involving probability distributions. In simple terms, the binomial coefficient refers to the number of ways a given event can occur. It is often notated as \( \binom{n}{k} \), which answers the question: 'In how many unique ways can we choose \( k \) elements from a larger set of \( n \) elements, regardless of the order of selection?'
When tossing a coin three times, as in the given exercise, the binomial coefficient helps us to calculate the number of ways to get a certain number of heads (or tails). For instance, to find the number of ways to get two heads (H) and one tail (T), we calculate \( \binom{3}{2} \), which equals 3 (HH(T), H(TH), (TH)H). Here each parenthesis indicates the position where a tail can be.
Remember, the formula to calculate the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n - k)!} \), where \( ! \) denotes factorial, which means multiplying a series of descending natural numbers (for example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)). Such calculations are key in determining the probability of events with multiple outcomes like in our coin toss scenario.

Independent Events

The concept of independent events is integral to solving problems involving probability. Independent events are scenarios where the outcome of one event does not influence or alter the outcome of another event. For instance, tossing a fair coin multiple times is an excellent example of independent events because the outcome of one toss does not affect the result of the next toss.
In our exercise, the three tosses of a coin are independent events. This means that getting heads or tails on the first toss does not change the probability of heads or tails on the subsequent tosses. Each toss still has a probability of \( 0.5 \) for heads and tails. As a result, if we want to calculate the probability of a specific sequence of events, we can multiply the probabilities of each event occurring. For example, the chance of getting three heads in a row is calculated by multiplying the probability of heads on each independent toss, which is \( 0.5 \times 0.5 \times 0.5 \) or \( 0.5^3 \).

Random Variable

A random variable is a numeric value that represents outcomes of a random phenomenon. In probability and statistics, random variables are used to quantify outcomes of random processes. They can be discrete, taking on a finite (or countably infinite) set of values, or continuous, taking on any value within an interval or collection of intervals.
In the given problem, \( X \) is defined as a discrete random variable representing the number of heads in the tosses minus the number of tails. The possible values \( X \) can take range from -3 to 3, including all integers in between. Here, we link each possible value of \( X \) with its corresponding probability, forming a probability distribution. This helps us understand the likelihood of various outcomes. For example, the probability \( p_{X}(k) \) indicates the likelihood of the random variable \( X \) being equal to a specific value \( k \). Calculating these probabilities involves using the binomial coefficient and understanding the independence of each event involved in the random process.