Question

If the coin is assumed fair, then, for \(n=2\), what are the probabilities associated with the values that \(X\) can take on?

Short answer

For \(n=2\) coin flips, the possible values of \(X\) (number of heads) are 0, 1, and 2. The associated probabilities are \(P(X = 0) = 1/4\), \(P(X = 1) = 1/2\), and \(P(X = 2) = 1/4\).

Step-by-step solution

Determine the possible outcomes for \(n=2\)

For \(n=2\), we will flip the coin twice. The possible outcomes are (H, H), (H, T), (T, H), and (T, T). Each of these outcomes has a probability of 1/4 (since the coin is fair).

Define the random variable \(X\) and find its values for the different outcomes

Let \(X\) be the number of heads obtained in the two coin flips. Now, we will determine the values of \(X\) for the different outcomes obtained in the first step: 1. (H, H): There are 2 heads, so \(X=2\). 2. (H, T): There is 1 head, so \(X=1\). 3. (T, H): There is 1 head, so \(X=1\). 4. (T, T): There are 0 heads, so \(X=0\). Now we can see that the possible values for \(X\) are 0, 1, and 2.

Calculate the probabilities associated with the values of \(X\)

We will now calculate the probabilities for each value of \(X\): 1. \(X = 0\): There is only one outcome with 0 heads (T, T), and its probability is 1/4. So, \(P(X = 0) = 1/4\). 2. \(X = 1\): There are two outcomes with 1 head - (H, T) and (T, H). Each has a probability of 1/4, so the total probability is 1/4 + 1/4 = 2/4 = 1/2. Hence, \(P(X = 1) = 1/2\). 3. \(X = 2\): There is only one outcome with 2 heads (H, H), and its probability is 1/4. So, \(P(X = 2) = 1/4\). To summarize, the probabilities associated with the values that \(X\) can take on for \(n=2\) are: - \(P(X = 0) = 1/4\) - \(P(X = 1) = 1/2\) - \(P(X = 2) = 1/4\)

Key concepts

Probability Theory

Probability theory is the mathematical framework that allows us to analyze random events and the likelihood of these events occurring. When flipping a fair coin, the outcome can only be heads (H) or tails (T), each with an equal probability. Since the coin is fair, we assume a 50% chance, or a probability of 0.5, for landing on either side.

In a sequence of coin flips, each flip is independent, meaning the outcome of one flip does not affect the next. In our exercise, flipping a coin twice (=2) leads to four possible combinations: (H, H), (H, T), (T, H), and (T, T). We calculated each of these outcomes to have an equal probability of 1/4 because the coin is fair and the likelihood of each sequence is equal.

In essence, probability theory helps us quantify uncertainty. By understanding the different possible outcomes and calculating their probabilities, we can predict the frequency of various results over many trials, a foundational principle in statistics and data analysis.

Random Variables

Random variables are a core concept in probability and statistics representing the potential outcomes of a random process. They are typically denoted by capital letters, such as \(X\) in our exercise.

A random variable can take on various values based on the occurrence of certain random events, with each value having an associated probability. For the coin flip example, let \(X\) be the number of heads obtained when flipping a coin twice. The random variable \(X\) can take on values 0, 1, or 2, corresponding to the number of heads that might occur. For instance, \(X=0\) represents no heads (T, T), \(X=1\) represents one head (either (H, T) or (T, H)), and \(X=2\) represents two heads (H, H).

By mapping outcomes to numerical values, random variables allow us to perform mathematical operations and calculations that enhance our understanding of chance and support decision-making based on probabilities.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood of a variable, which represents the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In the context of coin flipping, a success might be defined as flipping a head.

For our exercise with two coin flips, we are looking at a binomial distribution with \(n=2\) trials and a probability \(p=0.5\) of success (getting a head) on each trial. The binomial formula for calculating the probability of getting \(k\) heads in \(n\) flips is: \[ P(X = k) = \binom{n}{k}p^k(1-p)^{n-k} \], where \(\binom{n}{k}\) is the binomial coefficient. This formula tells us that the probability of obtaining exactly one head in two flips (\(X = 1\)), for example, is \[ P(X = 1) = \binom{2}{1}(0.5)^1(1-0.5)^{2-1} = 1/2 \].

Therefore, understanding the binomial distribution enables us to determine the probabilities of different numbers of 'successes' in a given number of trials – a vital concept for comprehending a variety of real-world phenomena and guiding experimental design.