The table below shows the score by some learners in basic 8. a. If the mean of the data is 7, find the value of a. b. find the mode c. find the probability of selecting a learner who scored the mode marks.

The mean is given by the formula:

From the table:
Scores ($x$): $5, 6, 7, 8, 9, 10$
Frequencies ($f$): $2, a, 4, 1, 2, 1$

Calculate $\sum (f \times x)$:

Calculate $\sum f$:

Given that the mean is 7:

Multiply both sides by $(10 + a)$:

Subtract $6a$ from both sides:

Subtract 70 from both sides:

The value of $a$ is $\boxed{\text{4}}$.

Scores ($x$): $5, 6, 7, 8, 9, 10$
Frequencies ($f$): $2, 4, 4, 1, 2, 1$
Total number of learners ($\sum f$) = $2 + 4 + 4 + 1 + 2 + 1 = 14$.

The mode is the score with the highest frequency.
The highest frequency in the updated data is 4, which corresponds to scores 6 and 7.
The mode is $\boxed{\text{6 and 7}}$.

The median is the middle value when the data is ordered. Since there are 14 learners (an even number), the median is the average of the $(\frac{14}{2})^{th}$ and $(\frac{14}{2} + 1)^{th}$ values, which are the $7^{th}$ and $8^{th}$ values.
Let's find the cumulative frequencies:

• Scores of 5: 2 learners (1st to 2nd position)
• Scores of 6: 4 learners (3rd to 6th position)
• Scores of 7: 4 learners (7th to 10th position) The $7^{th}$ value is 7, and the $8^{th}$ value is 7.

The median is $\boxed{\text{7}}$.

From part b), the mode marks are 6 and 7.
Number of learners who scored 6 marks = 4.
Number of learners who scored 7 marks = 4.
Total number of learners who scored mode marks = $4 + 4 = 8$.

The total number of learners is 14.
The probability of selecting a learner who scored the mode marks is:

Simplify the fraction:

The probability is $\boxed{\frac{4}{7}}$.