Question

If the table below represents a probability model based on observed frequencies, what is the value of x?

This table shows a probability model for the letters, Y, C, A, R, E, and S based on observed frequencies.
Y = P(Y)
C = 0.25
A = 0.18
R = 0.30
E = 0.17
S = x

a. 0.20
b. 0.10
c. 1
d. 0.01

Answer 1

To find the value of \( x \), we need to remember that the total probability must equal 1. This means we need to sum up the probabilities of all the letters and set that equal to 1.

From the table, we have:

- \( P(Y) = Y \)
- \( P(C) = 0.25 \)
- \( P(A) = 0.18 \)
- \( P(R) = 0.30 \)
- \( P(E) = 0.17 \)
- \( P(S) = x \)

Now we can sum these probabilities:

\[
Y + 0.25 + 0.18 + 0.30 + 0.17 + x = 1
\]

First, let's calculate the sum of the known probabilities:

\[
0.25 + 0.18 + 0.30 + 0.17 = 0.90
\]

Now we can substitute this back into the equation:

\[
Y + 0.90 + x = 1
\]

To isolate \( Y + x \), we need to move \( 0.90 \) to the other side:

\[
Y + x = 1 - 0.90
\]

This simplifies to:

\[
Y + x = 0.10
\]

Since the letter Y represents a probability itself, for \( Y \) to be a valid probability, it must be between \( 0 \) and \( 1 \).

If we assume \( Y = 0 \) (which is the maximum possible value for \( x \)), then:

\[
0 + x = 0.10 \implies x = 0.10
\]

If \( Y \) were any positive value, then \( x \) would have to decrease correspondingly while still keeping the sum equal to \( 0.10 \).

Since we want to find the value of \( x \) that satisfies \( Y + 0.90 + x = 1 \) and remains within the defined probabilities:

Thus, \( x = 0.10 \) is a valid solution that ensures the entire sum adds up correctly:

- So we can conclude that the value of \( x \) is

**b. 0.10**