Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 775 — #783
18.9. Probability versus Confidence 775
This shows that after flipping one hundred heads, the odds that the biased coin was
chosen are overwhelming, and so with high probability the next flip will be a Head.
Thus, by assuming some tiny probability for the coin being heavily biased toward
Heads, we can justify our intuition that after one hundred consecutive Heads, the
next flip is very likely to be a Head.
Making an assumption about the probability that some unverified fact is true is
known as the Bayesian approach to a hypthesis testing problem. By granting a tiny
probability that the biased coin was being flipped, this Bayesian approach provided
a reasonable justification for estimating that the odds of a Head on the next flip are
ninety-nine to one in favor.
18.9.5 Confidence in the Next Flip
If we stick to confidence rather than probability, we don’t need to make any Bayesian
assumptions about the probability of a fair coin. We know that if one hundred
Heads are flipped, then either the coin is biased, or something that virtually never
happens (probability 2 100 ) has occurred. That means we can assert that the coin
is biased at the 1 2 100 confidence level. In short, when one hundred Heads are
flipped, we can be essentially 100% confident that the coin is biased.
Problems for Section 18.4
Homework Problems
Problem 18.1.
The Conditional Probability Product Rule for n Events is
Rule.
PrŒE 1 \ E 2 \ : : : \ E n D PrŒE 1 Pr E 2 j E 1
Pr
E3 j E 1 \ E 2
(a) Restate the Rule without using elipses (. . . ).
(b) Prove it by induction.
Pr E n j E 1 \ E 2 \ : : : \ E n 1
: