Question

Use the probability distribution for the random variable \(x\) to answer the questions. $$\begin{array}{l|lllll}x & 0 & 1 & 2 & 3 & 4 \\\\\hline p(x) & .1 & .3 & .3 & ? & .1\end{array}$$ What is the probability that \(x\) is greater than \(2 ?\)

Short answer

Answer: The probability that \(x\) is greater than 2 is \(0.2\).

Step-by-step solution

Find the missing probability

Since the sum of all probabilities in a probability distribution must equal to 1, by adding all the given probability values, and subtracting from 1, we will find the missing probability. Missing probability = \(1 - (p(0) + p(1) + p(2) + p(4)) = 1 - (0.1 + 0.3 + 0.3 + 0.1) = 0.1\) So, the probability distribution is: $$\begin{array}{l|lllll}x & 0 & 1 & 2 & 3 & 4 \\\\\hline p(x) & .1 & .3 & .3 & 0.1 & .1\end{array}$$

Calculate the probability that \(x\) is greater than \(2\)

To calculate the probability that \(x\) is greater than 2, we need to sum the probabilities of all values greater than 2. The possible values of \(x\) greater than 2 are \(3\) and \(4\). So, we add the probabilities for \(p(3)\) and \(p(4)\). \(P(x>2) = p(3) + p(4) = 0.1 + 0.1 = 0.2\) The probability that \(x\) is greater than \(2\) is \(0.2\).

Key concepts

Random Variable

In the realm of probability and statistics, a random variable is a fundamental concept which represents a numerical outcome of a random phenomenon. Imagine rolling a die; the result of the toss – any number from 1 to 6 – is a random variable. In our context, the random variable is not a die, but it represents the possible outcomes for the variable x, which has a set of predetermined possibilities, such as 0, 1, 2, 3, and 4.

Each possible outcome of a random variable is associated with a probability, signifying how likely that specific outcome is to occur. The way these probabilities are assigned to each outcome is through what we call a probability distribution. This distribution is visualized as a table showing each outcome along with the respective probability, often denoted as p(x) where x is a random variable outcome.

Probability Calculation

To establish a clear understanding of probability calculation, let's delve into how exactly probabilities are computed in the context of a random variable. Probabilities are numerical values that range from 0 to 1, inclusive. A probability of 0 denotes an impossible event, while a probability of 1 indicates certainty.

When a probability distribution is given, calculation often involves adding up the probabilities of the desired outcomes. For instance, if we seek to identify the probability of the random variable x taking on a value greater than 2, we simply accumulate the probabilities assigned to outcomes 3 and 4. It's akin to collecting pieces that fit the condition and summing their probabilities to find the total like in our exercise:

• Add p(3) and p(4) to compute P(x > 2).

This sum represents the probability of the event where the random variable x is greater than 2.

Sum of Probabilities

One of the foundational principles in probability theory is the sum of probabilities, which states that the probabilities of all possible outcomes should add up to 1. Why is this the case? Because the sum expresses the certainty that one of the possible outcomes will occur. When we talk about a complete set of outcomes, we mean that no other outcome is possible.

So when the problem at hand contains a missing probability, like our unknown p(3), we use this principle to find it. We sum the probabilities of the known outcomes and subtract from 1, revealing that missing piece. It's much like putting together a puzzle where you already know the total picture and just need to fit in the missing piece. This ensures that our entire set of probable outcomes perfectly aligns with the established rule that they must sum to 1.