Directions (55-57): Read the following table carefully and answer the questions given below.

In the table some numbers are given and the probability of picking that number is given.

NUMBERS	Probability
Q	$\frac{1}{6}$
2.5	$\frac{1}{3}$
Q − 2.5	$\frac{1}{3}$
P	$\frac{1}{6}$
$\frac{4Q-5}{2}$	$\frac{1}{6}$
18.75	$\frac{1}{6}$

If R = 5Q – 20, then find the value of R.

Step 1: Understand the properties of a probability distribution

For any valid probability distribution, the sum of the probabilities of all unique (distinct) outcomes must be equal to 1. Let's analyze the given table of numbers and their respective probabilities:

Step 2: Solve for Q

If we sum all the probabilities listed in the table directly, we get:
$\frac{1}{6}+\frac{1}{3}+\frac{1}{3}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{8}{6}=\frac{4}{3}$
Since the total probability cannot exceed 1, this indicates that some of the listed numbers must represent the same value, thereby overlapping and preventing double-counting.

To reduce the sum of probabilities to exactly 1, we must subtract $\frac{1}{3}$ from the total sum of $\frac{4}{3}$ (since $\frac{4}{3}-\frac{1}{3}=1$). This implies that the outcome with value $Q-2.5$ must be identical to the outcome with value $2.5$.

Setting these two equal, we have:
$Q-2.5=2.5$
Adding 2.5 to both sides gives:
$Q=5$

Step 3: Calculate the value of R

We are given the relation:
$R=5Q-20$

Substitute the value of $Q=5$ into the equation:
$R=5\left(5\right)-20$
$R=25-20$
$R=5$

Thus, the value of R is 5.