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A certain mathematics interest group conducted a study on the relationship between the mathematics and Chinese language scores of students in their school. They used a simple random sampling method with replacement to draw a sample of size $200$ from the school. The sample observation data of mathematics and Chinese language scores are organized as follows:

|          | Chinese Score |         |         | Total |
|----------|---------------|---------|---------|-------|
| Math Score | Excellent     | Not Excellent |         |       |
| Excellent | 50            | 30      | 80      |
| Not Excellent | 40         | 80      | 120     |
| Total    | 90            | 110     | 200     |

$(1)$ Based on an independence test with $\alpha = 0.010$, can it be concluded that there is an association between mathematics scores and Chinese language scores?  
$(2)$ In artificial intelligence, $L({B|A})=\frac{{P({B|A})}}{{P({\overline{B}|A})}}$ is commonly used to represent the odds of event $B$ occurring given that event $A$ has occurred, known as the likelihood ratio in statistics. Now, randomly select a student from the school, where $A$ represents "the selected student has a non-excellent Chinese language score" and $B$ represents "the selected student has a non-excellent mathematics score." Using the sample data, estimate the value of $L(B|A)$.  
$(3)$ Now, using stratified sampling, select $8$ students from the sample of students with excellent mathematics scores. From these $8$ students, randomly select $3$ students to participate in a mathematics competition. Determine the probability distribution table of the number $X$ of students with excellent Chinese language scores among these $3$ students and calculate the mathematical expectation.  
Given: ${\chi^2}=\frac{{n{{(ad-bc)}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$.  
| $\alpha$ | 0.050 | 0.010 | 0.001 |
|----------|-------|-------|-------|
| $x_{\alpha}$ | 3.841 | 6.635 | 10.828 |

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