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. $$ \begin{array}{lccc} \hline x & 1 & 2 & 3 \\ \hline p(x) & 0.5 & 0.6 & ? \\ \hline \end{array} $$

Short answer

It is not possible to make \(p(x)\) a probability function with the given probabilities as the calculated probability for \(x=3\) is -0.1, which falls out of the acceptable range (0 to 1) for probabilities.

Step-by-step solution

Identifying given probabilities

Given that the probability \(p(x)\) for \(x=1\) and \(x=2\) are 0.5 and 0.6 respectively.

Formulate the equation to find the unknown probability

Since it is a probability function, the sum of all probabilities should be congruent to 1. So, the equation to solve for the unknown probability for \(x=3\) would be \(p(1) + p(2) + p(3) = 1 \), which translates to \(0.5 + 0.6 + p(3) = 1\).

Solving for the unknown probability

To find the value of \(p(3)\), subtract the sum of the known probabilities from 1: \(p(3) = 1 - (0.5 + 0.6) = 1 - 1.1 = -0.1 \)

Check if the solution is valid

The probability of an event are always between 0 and 1, inclusive. However, \(p(3) = -0.1\) which falls out of this range. Therefore, it is not possible to make \(p(x)\) a probability function under the given conditions.