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. $$ \begin{array}{lcccc} \hline x & 10 & 20 & 30 & 40 \\ \hline p(x) & 0.2 & 0.2 & ? & 0.2 \\ \hline \end{array} $$

Short answer

The missing probability value at x=30 should be 0.4 to make \(p(x)\) a valid probability function.

Step-by-step solution

Sum the known probabilities

Firstly, sum up all known probabilities given in the table, which include 0.2 at x=10, 0.2 at x=20, and 0.2 at x=40 which totals to 0.6.

Calculate the unknown probability

Since the total probability must equal to 1 under the axiom of probability, you subtract the sum of known probabilities from 1 to find the missing probability:\n 1 - 0.6 = 0.4\n Therefore, the missing probability value for x=30 is 0.4.

Check the conditions of probability

Finally, check the two conditions of probability with this filled value. Here, the computed probability value for x=30 is not negative, so it satisfies the 1st condition. Moreover, if we add up all probabilities in the table now, it equals to 1 which satisfies the 2nd condition. Therefore, these values make \(p(x)\) a valid probability function.