Problem 18 - A First Course in Probability

Question

Four independent flips of a fair coin are made. Let \(X\) denote the number of heads obtained. Plot the probability mass function of the random variable \(x-2\)

Step-by-Step Solution

Answer

To plot the probability mass function (pmf) of the random variable \(X-2\), where \(X\) denotes the number of heads obtained in four independent flips of a fair coin, first list the possible outcomes and their probabilities: - For \(Y=-2\) (0 heads), probability is 1/16. - For \(Y=-1\) (1 head), probability is 4/16. - For \(Y=0\) (2 heads), probability is 6/16. - For \(Y=1\) (3 heads), probability is 4/16. - For \(Y=2\) (4 heads), probability is 1/16. Plot these points on a graph with the x-axis representing the possible values of \(Y\) and the y-axis representing the corresponding probabilities: - (-2, 1/16) - (-1, 4/16) - (0, 6/16) - (1, 4/16) - (2, 1/16)

Step 1: Possible Outcomes

List down the possible outcomes of flipping a fair coin 4 times. There can be 0, 1, 2, 3, or 4 heads obtained.

Step 2: Probabilities of Outcomes

Next, determine the probabilities of each outcome. Since each coin flip has 2 possible outcomes (heads or tails), with equal probability of 1/2, there are a total of \(2^4\) = 16 possible outcomes of flipping a fair coin 4 times. Here's a breakdown of the probabilities for each number of heads obtained: 0 heads: There is 1 way to get 0 heads (all tails), so the probability is 1/16. 1 head: There are 4 ways to get 1 head (one in each toss), so the probability is 4/16. 2 heads: There are 6 ways to get 2 heads, so the probability is 6/16. 3 heads: There are 4 ways to get 3 heads, so the probability is 4/16. 4 heads: There is 1 way to get 4 heads (all heads), so the probability is 1/16.

Step 3: pmf of the Random Variable X - 2

Now, we need to find the probability mass function (pmf) of the random variable \(X - 2\). Let \(Y = X - 2\). We will now compute the pmf of \(Y\) for each possible value. Possible values of \(Y\) are \(-2, -1, 0, 1, 2\), corresponding to the different number of heads obtained. Because \(Y = X - 2\), we can rewrite the probabilities in terms of \(Y\): - For \(Y = -2\) (0 heads), probability is 1/16. - For \(Y = -1\) (1 head), probability is 4/16. - For \(Y = 0\) (2 heads), probability is 6/16. - For \(Y = 1\) (3 heads), probability is 4/16. - For \(Y = 2\) (4 heads), probability is 1/16.

Step 4: Plotting the pmf of Random Variable X - 2

Finally, plot the pmf of random variable \(X-2\) on the graph where the x-axis represents the possible values of \(Y\) and the y-axis represents the corresponding probabilities. The plot will include the following points: - (-2, 1/16) - (-1, 4/16) - (0, 6/16) - (1, 4/16) - (2, 1/16)

Key Concepts

Random Variable

Imagine a random variable as a tool used to predict possible outcomes of a random phenomenon, like flipping a coin. In our case, the random variable is the number of heads we get when a fair coin is flipped four times. We call this random variable \(X\).
\(X\) can take on any integer value between 0 and 4 because it's possible to get anywhere from zero heads (all tails) to four heads (all heads). This range is known as the support or set of possible values the random variable can take.
Random variables help us by translating real-world scenarios into a mathematical framework. When we shift \(X\) by subtracting 2, we get a new random variable \(Y = X - 2\). It still represents the number of heads, just adjusted by 2 which shifts the support to \(-2, -1, 0, 1, 2\). This means the way we think about possible outcomes just gets moved over on the graph.

Binomial Distribution

Binomial distribution is like the magic behind understanding the flip of a coin multiple times. When you flip a coin four times, there’s a pattern hiding in the outcomes called the binomial distribution.
This magical formula considers two possibilities: heads or tails. In mathematics, a binomial distribution is given by the probabilities of obtaining a certain number of heads in several coin flips. With our four flip example, each flip is independent. So, the probability of getting heads remains \(\frac{1}{2}\).

• 0 heads happens in 1 way: all flips are tails, so the probability is \(\frac{1}{16}\).
• 1 head appears in 4 ways, like when any one of the four flips results in a head, making its probability \(\frac{4}{16}\).
• 2 heads come in 6 ways. Its probability is \(\frac{6}{16}\), calculated using combinations.
• 3 heads also comes in 4 ways, with a \(\frac{4}{16}\) probability.
• 4 heads can only occur 1 way: all flips are heads, giving a probability of \(\frac{1}{16}\).

Recognizing this pattern teaches us how to predict outcomes not just for coins, but for any event where there are two possible initial outcomes.

Discrete Probability

Discrete probability deals with events that have a finite number of possible outcomes. When we're flipping four coins, we only have a limited number (5 possible outcomes), all linked to the number of heads that could appear.
The probabilities we see, like \(\frac{6}{16}\) for getting 2 heads, are specific outcomes from our discrete set of possibilities. This means we're not guessing, but effectively calculating exact likelihoods.
To visualize this, we use a probability mass function (pmf). The pmf is like a map of these probabilities. For example, the value \(Y = 0\) (or 2 heads) is represented as \(\frac{6}{16}\) in the graph. Each bar on the plot corresponds to how likely each of these outcomes is. This visual tool helps us understand and compare the discrete probabilities at a glance, making predictions about real-world scenarios more manageable.