The same 15 contestants, on each of 3 days, answered 5 questions in order to win a prize. Each contestant received 1 point for each correct answer. The number of contestants receiving a given score on each day is shown in the table below.

$$ \text{Number of Contestants by Score and Day} $$ $$ \small \begin{array} {|c|c|c|c|c|c|c|c|} \hline & 5\text{ out} & 4\text{ out} &3\text{ out} &2\text{ out} & 1\text{ out} & 0\text{ out} \\ & \text{of } 5 & \text{of } 5 & \text{of } 5 & \text{of } 5 & \text{of } 5 & \text{of } 5 & \text{Total} \\ \hline \text{Day 1} & 1 & 2 & 3 & 5 & 2 & 2 & 15 \\ \hline \text{Day 2} & 3 & 2 & 2 & 2 & 3 & 3 & 15 \\ \hline \text{Day 3} & 1 & 7 & 3 & 1 & 1 & 2 & 15 \\ \hline \text{Total} & 5 & 11 & 8 & 8 & 6 & 7 & 45 \\ \hline \end{array} $$

No contestant received the same score on two different days. If a contestant is selected at random, what is the probability that the selected contestant received a score of 3 on Day 1 or Day 2, given that the contestant received a score of 3 on one of the three days?

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We are looking for the following:

$$ \frac{\text{score of 3 on day 1 or 2}}{\text{score of 3}} $$

Since no contestant scored a score more than once, there are 8 contestants total who scored a score of 3.

$$ \text{Number of Contestants by Score and Day} $$ $$ \begin{array} {|c|c|c|c|c|c|c|c|} \hline & 5\text{ out} & 4\text{ out} &3\text{ out} &2\text{ out} & 1\text{ out} & 0\text{ out} \\ & \text{of } 5 & \text{of } 5 & \text{of } 5 & \text{of } 5 & \text{of } 5 & \text{of } 5 & \text{Total} \\ \hline \text{Day 1} & 1 & 2 & \colorbox{yellow}{$3$} & 5 & 2 & 2 & 15 \\ \hline \text{Day 2} & 3 & 2 & \colorbox{yellow}{$2$} & 2 & 3 & 3 & 15 \\ \hline \text{Day 3} & 1 & 7 & 3 & 1 & 1 & 2 & 15 \\ \hline \text{Total} & 5 & 11 & \colorbox{aqua}{$8$} & 8 & 6 & 7 & 45 \\ \hline \end{array} $$ $$ \frac{\text{score of 3 on day 1 or 2}}{\text{score of 3}} $$ $$ = \frac{3+2}{8} $$ $$ = \boxed{\frac{5}{8}} $$