To find the value of [tex]\( p \)[/tex] given that the probabilities of the three outcomes are [tex]\( p \)[/tex], [tex]\( \frac{p}{2} \)[/tex], and [tex]\( \frac{p}{4} \)[/tex], we need to use the fact that the sum of the probabilities of all possible outcomes must be equal to 1. This can be set up as the following equation:
[tex]\[
p + \frac{p}{2} + \frac{p}{4} = 1
\][/tex]
First, we need to find a common denominator for the terms on the left-hand side of the equation. The common denominator for 1, 2, and 4 is 4. Rewrite each term with this common denominator:
[tex]\[
p = \frac{4p}{4}, \quad \frac{p}{2} = \frac{2p}{4}, \quad \frac{p}{4} = \frac{p}{4}
\][/tex]
Now, substitute these expressions back into the equation:
[tex]\[
\frac{4p}{4} + \frac{2p}{4} + \frac{p}{4} = 1
\][/tex]
Combine the fractions:
[tex]\[
\frac{4p + 2p + p}{4} = 1
\][/tex]
Simplify the numerator:
[tex]\[
\frac{7p}{4} = 1
\][/tex]
To solve for [tex]\( p \)[/tex], multiply both sides of the equation by 4:
[tex]\[
7p = 4
\][/tex]
Now, divide both sides by 7:
[tex]\[
p = \frac{4}{7}
\][/tex]
Thus, the value of [tex]\( p \)[/tex] is:
[tex]\[
\boxed{\frac{4}{7}}
\][/tex]
