Q1.
If I have rolled a 4 on a die 8 times in succession, the probability of rolling a 4 again is still only ≈ 0.16 (assuming the die is fair). Probability does not tell you what will happen next but predicts what will happen in the long run.
Answer
Correct — and worth stating exactly. For a fair die the next roll is a fresh experiment with sample space S = {1, 2, 3, 4, 5, 6}, so
P(next roll is a 4) = 1/6 = 0.1666… ≈ 0.167
(the book rounds this to ≈ 0.16)
(the book rounds this to ≈ 0.16)
Why it happens: the die carries no record of the eight rolls that went before. Each roll is an independent event, so the sample space and the favourable count are the same as they were on the very first roll. There is no mechanism by which a past outcome could change the shape of a cube.
The trap is to confuse two different questions:
- Before any rolling: P(nine 4s in a row) = (1/6)9 ≈ 0.0000001 — a genuinely tiny number.
- After eight 4s have already happened: those eight are now facts, not possibilities. Only the ninth roll is still uncertain, and for it P = 1/6.
The rare thing is the whole run of nine, not the last roll of it. Believing that a 4 has become "unlikely now" — or, in the other direction, that the die is "hot" — is the Gambler's Fallacy of page 164.
Check it yourself: there is a sensible worry hiding here, but it is a different one. Eight 4s in a row is such weak evidence for a fair die that you might reasonably start doubting the assumption and test whether the die is loaded. That is a question about the die, not about the ninth roll. As long as the die is fair, 1/6 stands.