First, let's solve for the probability that any given game number has exactly 3 out of 5 days with the same number. That answer is COMBIN(5,3)*(1/80)^2*(79/80)^2 = 0.001523682. This is the number of ways to choose the 3 matching days out of 5 times the probability the second and third day match the first times the probability the other two days don't.
So, the probability there is not a 3 out of 5 day match for any given game number is 1 - 0.001523682 = 0.9984763.
The probability this does not happen for 300 days is 0.9984763300 = 63.29%.
Thus, the probability of the alternative, that there is at least one drawing number with 3 out of 5 days matching the same Bulls Eye number is 36.71%.
This question is asked and discussed in my forum at Wizard of Vegas.