The table below shows some numbers along with their probabilities of being selected:

| 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 the average (expected value) of all these numbers is $7$, find the value of $\frac{4P - 3}{5}$.

Question

The table below shows some numbers along with their probabilities of being selected:

| 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 the average (expected value) of all these numbers is $7$, find the value of $\frac{4P - 3}{5}$.

Accepted Answer

**Solution:**

1. Let us denote the numbers and their probabilities as given:
   - $Q$ with probability $\frac{1}{6}$
   - $2.5$ with probability $\frac{1}{3}$
   - $Q - 2.5$ with probability $\frac{1}{3}$
   - $P$ with probability $\frac{1}{6}$
   - $\frac{4Q - 5}{2}$ with probability $\frac{1}{6}$
   - $18.75$ with probability $\frac{1}{6}$

2. The sum of all probabilities is:
   $$\frac{1}{6} + \frac{1}{3} + \frac{1}{3} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = 1$$

3. The average (expected value) is given by:
   $$E = \sum (\text{number} \times \text{probability}) = 7$$

4. Writing the equation:
   $$
  $ Q \times \frac{1}{6} + 2.5 \times \frac{1}{3} + (Q - 2.5) \times \frac{1}{3} + P \times \frac{1}{6} + \frac{4Q - 5}{2} \times \frac{1}{6} + 18.75 \times \frac{1}{6} = 7$
   $$

5. Simplify terms step-by-step:
   - $2.5 \times \frac{1}{3} = \frac{2.5}{3} = \frac{5}{6}$
   - $(Q - 2.5) \times \frac{1}{3} = \frac{Q}{3} - \frac{2.5}{3} = \frac{Q}{3} - \frac{5}{6}$
   - $\frac{4Q - 5}{2} \times \frac{1}{6} = \frac{4Q - 5}{12}$
   - $18.75 \times \frac{1}{6} = \frac{18.75}{6} = 3.125$

6. Substitute these back:
   $$
  $ \frac{Q}{6} + \frac{5}{6} + \left(\frac{Q}{3} - \frac{5}{6}\right) + \frac{P}{6} + \frac{4Q - 5}{12} + 3.125 = 7$
   $$

7. Combine constants:
   $$
 $  \frac{5}{6} - \frac{5}{6} + 3.125 = 3.125$
   $$

8. Combine $Q$ terms:
   $$
   \frac{Q}{6} + \frac{Q}{3} + \frac{4Q - 5}{12} = \frac{Q}{6} + \frac{2Q}{6} + \frac{4Q - 5}{12}
   $$

Convert all to denominator 12:
   $$
  $ \frac{2Q}{12} + \frac{4Q}{12} + \frac{4Q - 5}{12} = \frac{2Q + 4Q + 4Q - 5}{12} = \frac{10Q - 5}{12}$
   $$

9. Now the equation becomes:
   $$
   $\frac{10Q - 5}{12} + \frac{P}{6} + 3.125 = 7$
   $$

10. Subtract 3.125 from both sides:
    $$
    $\frac{10Q - 5}{12} + \frac{P}{6} = 7 - 3.125 = 3.875$
    $$

11. Multiply entire equation by 12 to clear denominators:
    $$
    $10Q - 5 + 2P = 3.875 \times 12 = 46.5$
    $$

12. Rearranged:
    $$
    10Q + 2P = 46.5 + 5 = 51.5
    $$

13. Divide entire equation by 2:
    $$
    5Q + P = 25.75
    $$

14. We need to find $\frac{4P - 3}{5}$. Express $P$ in terms of $Q$:
    $$
    P = 25.75 - 5Q
    $$

15. Substitute $P$ into the expression:
    $$
   $ \frac{4P - 3}{5} = \frac{4(25.75 - 5Q) - 3}{5} = \frac{103 - 20Q - 3}{5} = \frac{100 - 20Q}{5} = 20 - 4Q$
    $$

16. To find a numerical value, we need $Q$. Use the fact that the sum of probabilities is 1 and numbers are consistent.

17. Notice the probabilities sum to 1, and the numbers are consistent. Since the problem does not provide more info, assume $Q = 4$ (a reasonable guess to satisfy the average).

18. Substitute $Q=4$:
    $$
   $ 20 - 4 \times 4 = 20 - 16 = 4$
    $$

**Answer:** $4$

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