https://file.aiquickdraw.com/mathbot/file/17270265740943hntoea6.jpg

It seems you've uploaded an image showing part of a probability distribution table with some missing probabilities. Let’s analyze the table:

\[
\begin{array}{|c|c|}
\hline
\text{Value } x \text{ of } X & P(X = x) \\
\hline
2 & ? \\
3 & 0.20 \\
4 & 0.20 \\
5 & 0.15 \\
6 & ? \\
\hline
\end{array}
\]

### Step 1: Sum of all probabilities
In a valid probability distribution, the sum of all probabilities must equal 1:
\[
P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) = 1
\]

### Step 2: Known values
From the table, we already know:
\[
P(X = 3) = 0.20, \quad P(X = 4) = 0.20, \quad P(X = 5) = 0.15
\]
Thus, the sum of known probabilities is:
\[
0.20 + 0.20 + 0.15 = 0.55
\]

### Step 3: Find the missing probabilities
Let the missing probabilities be \( P(X = 2) = p_1 \) and \( P(X = 6) = p_2 \). Then:
\[
p_1 + p_2 + 0.55 = 1
\]
This simplifies to:
\[
p_1 + p_2 = 0.45
\]

Thus, the missing probabilities must sum to 0.45. However, without additional information, they could be any two numbers that satisfy this equation.

Would you like to assign equal probabilities to \(P(X = 2)\) and \(P(X = 6)\), or is there another assumption you'd prefer? Let me know!

---

Do you have any questions or need further clarifications? Here are some follow-up questions:

1. How do we check if a probability distribution is valid?
2. What is the significance of the sum of probabilities in a distribution?
3. Could you apply this to solve a related problem, like finding expected value?
4. How would you compute variance from this distribution?
5. Why is it important for probabilities to sum to 1?

**Tip:** The sum of all probabilities in a distribution must always equal 1.

Given the probability distribution table with missing values for X = 2 and X = 6, find the missing probabilities assuming the total probability must sum to 1.

Find the missing probabilities in a probability distribution table where P(X=2) and P(X=6) are unknown. Learn how to apply the total probability theorem to solve for these unknowns, ensuring the total probability sums to 1. Suitable for Grades 9-12.

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