Explanation:
the probability to win is 0.2.
the probability to lose is 1-02 = 0.8.
there is no sure win after at least n trials.
on day 1 the probability to win is 0.2.
on day 2 the probability to win at least once is
the probability to have won on day 1 but not on day 2, plus the probability to have won on day 2 but not on day 1 plus the probability to have won on both days :
0.2×0.8 + 0.8×0.2 + 0.2×0.2 = 0.32 + 0.04 = 0.36
...
on day 7 we have
the probability to have won on one day but not on 6 days times the combination of picking 1 day out of 7,
plus the probability to have won on 2 days but not on 5 days times the combinations to pick 2 out of 7,
plus the probability to have won on 3 days but not on 4 days times the combinations to pick 3 out of 7,
...
plus the probability to have won on all 7 days.
that is
0.2.×0.8⁶ × 7 +
+ 0.2²×0.8⁵ × C(7, 2) +
+ 0.2³×0.8⁴ × C(7, 3) +
+ 0.2⁴×0.8³ × C(7, 4) +
+ 0.2⁵×0.8² × C(7, 5) +
+ 0.2⁶×0.8 × 7 +
+ 0.2⁷
but before we lose ourselves in tons of little operations, this is the same as the opposite of having lost on all 7 days.
that is
1 - 0.8⁷ = 1 - 0.2097152 = 0.7902848 ≈ 0.7903
so, the probability that Jorge has won at least one prize is
0.7903