Probability?

Method 1 :
In rolling a die once :
Pr(1 to 4) = 4/6 = 2/3
Pr(5 or 6) = 2/6 = 1/3

Pr(roll the die fewer than 8 times)
= 1 - P(lost in the first 7 times)
= 1 - (2/3)⁷
= 1 - 128/2187
= 2059/2187


Method 2 :
Pr(roll 1 time) = 1/3
Pr(roll 2 times) = (2/3) × (1/3) = 2/9
Pr(roll 3 times) = (2/3)² × (1/3) = 4/27
Pr(roll 4 times) = (2/3)³ × (1/3) = 8/81
Pr(roll 5 times) = (2/3)⁴ × (1/3) = 16/243
Pr(roll 6 times) = (2/3)⁵ × (1/3) = 32/729
Pr(roll 7 times) = (2/3)⁶ × (1/3) = 64/2187

Pr(roll the die fewer than 8 times)
= Pr(roll 1 time) + Pr(roll 2 times) + ...... + Pr(roll 7 times)
= (1/3) + (2/9) + (4/27) + (8/81) + (16/243) + (32/729) + (64/2187)
= 2059/2187

p(5 or 6) = p(5) + p(6) = 1/6 + 1/6 = 2/6 = 1/3

(1/3)^7 = 1/2187

Answer: 99.954275%