91Ó°ÊÓ

A probability is a measure that quantifies the likelihood of an event occurring. Probabilities are expressed as numbers between 0 and 1, where 0 indicates an impossible event and 1 signifies a certain event.
To understand probability better, think of simple examples like flipping a coin or rolling a die. When you flip a coin, the probability of getting heads is 0.5, because there are two equally likely outcomes (heads or tails).
In mathematics, probabilities for events are often denoted by the function \( f(x) \), where \( x \) is a possible outcome. For example, if you're rolling a standard six-sided die, the probability of rolling a three can be denoted as \( f(3) \).

For a set of events that constitute a probability distribution, the total sum of the probabilities must be 1. This is a fundamental property of probability distributions.
Think of it as distributing 100% chance among all possible outcomes. If you're rolling a six-sided die, the sum of the probabilities for each outcome (1 through 6) must equal 1. Mathematically, if you have probabilities \(f(1), f(2), \ldots, f(6)\), then it follows:
\begin{itemize} \ \left( f(1) + f(2) + f(3) + f(4) + f(5) + f(6) = 1 \right)<. \end{itemize} < br> . In our exercise, we have the probabilities \( f(3) = 0.4 \), \( f(5) = 0.1 \), and \( f(6) = 0.2 \). To complete the distribution, the missing probability \( f(4) \) must be such that their sum totals to 1.

To find a missing probability in a discrete probability distribution, you need to follow a few steps:

• Sum up all the known probabilities.
• Subtract this sum from 1 to find the missing probability.

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In our example, the known probabilities are 0.4, 0.1, and 0.2. When we add these together, we get 0.7.
Now, we subtract this from 1 to find the missing probability:

• \begin{equation*} \text{Missing probability} = 1 - 0.7 = 0.3 \end{equation*}.

To confirm our calculation, we can check if the sum of all probabilities equals 1:

\begin{equation*} \left(0.4 + 0.3 + 0.1 + 0.2 = 1 \right) \end{equation*}.

Everything checks out, so \( f(4) = 0.3 \) completes the distribution.