Question

In Exercises \(\mathrm{P} .74\) to \(\mathrm{P} .77\), fill in the \(?\) to make \(p(x)\) a probability function. If not possible, say so. $$ \begin{array}{lcccc} \hline x & 1 & 2 & 3 & 4 \\ \hline p(x) & 0.1 & 0.1 & 0.2 & ? \\ \hline \end{array} $$

Short answer

The missing probability for the value 4 is 0.6.

Step-by-step solution

Identify given probabilities

The given probabilities for the values 1, 2, and 3 are 0.1, 0.1, and 0.2 respectively.

Sum the given probabilities

The sum of the given probabilities is \(0.1 + 0.1 + 0.2 = 0.4\).

Find the missing probability

We know that the total probability is 1. Therefore, the missing probability for the value 4 can be found by subtracting the sum of the given probabilities from 1. So, the missing probability is : \(1 - 0.4 = 0.6\)

Key concepts

Total Probability

When delving into the realm of probability, it is fundamental to understand the concept of total probability. It is a principle stating that the sum of probabilities for all possible outcomes of a random experiment must equal 1. This is because the probability of some outcome occurring is certain when considering all possible outcomes.

For example, when we flip a fair coin, there are only two possible outcomes: heads or tails. The probability of getting heads might be 0.5, and the probability of getting tails is also 0.5. When we add these probabilities together, we get 1, which aligns with our certainty that the coin will land on either heads or tails.

In the context of the given exercise, the total probability helps us to solve for the missing value of a probability function. By knowing that the total probability must equal 1, the missing probability can be found by subtracting the sum of the known probabilities from 1. This approach is used in step 3 of the textbook's solution to determine the value of the unknown probability.

Probability Distribution

A probability distribution is a mathematical description of the likelihood of various outcomes in an experiment. It can take various forms, such as a table, formula, or graph, but its fundamental purpose is to map out the probabilities of all possible outcomes.

In a discrete probability distribution, like the one presented in our exercise, each possible outcome of a random variable is listed along with its corresponding probability. It's important to ensure that the probabilities for all potential outcomes are non-negative and that their sum equals 1, adhering to the rules of total probability.

The provided exercise showcases a simple probability distribution table for a random variable taking on values 1 through 4. To complete the distribution, we use the rule that the sum of all probabilities must equal 1, thus helping us find the missing value needed to make the table represent a valid probability distribution.

Sum of Probabilities

The sum of probabilities concept is a cornerstone in the study of probability. It implies that when dealing with a finite number of discrete outcomes, the probabilities associated with these outcomes must add up to 1, which is a representation of certainty that one of the outcomes will occur.

As we see in our exercise, the given probabilities for three of the outcomes are added together to yield a partial sum. This sum will always be less than or equal to 1 because no single outcome can have a probability greater than 1, and there must be room for all other possible outcomes.

The importance of this principle cannot be overstated in practice. It ensures that when we calculate the probability of a specific set of outcomes or when searching for an unknown probability in a function, our calculations reflect a complete and accurate model of the real-world scenario being analyzed. This is precisely the methodology applied in solving the textbook problem, where the sum of known probabilities (0.1 + 0.1 + 0.2) is subtracted from 1 to find the missing probability.