Note:   throughout ch(P) = the chance of P; cr(P) = your credence in P;

1. Suppose that Law is the proposition which encapsulates the truth about the laws of nature. A miracle is any proposition P such that Law entails that P has a very low chance of being true: that is, ch(P)≤1/2¹⁰⁰.  Assume the following:

(a) cr(Laws)= 2¹⁰⁰/(2¹⁰⁰+1).  (You have a very high credence in Law.)

(b) Law entails ch(M)=1/2¹⁰¹.  (M would indeed be a miracle given that Law is true.)

(c) Suppose you are agnostic about whether M would occur if Law were false.  That is: cr(M|¬Law) = 1/2.

(d)  The Principal Principle:  if Q entails the proposition ch(P)=r  then cr(P |Q)=r.

Questions:

1. Evaluate: cr(M|Law).
2. Evaluate: cr(¬Law).

• Using your answers to (i) and (ii), (b), (c) and the Law of Total Probability, show that:

ch(M)  < cr(M) ≤ cr(¬Law).

2    C^r is the hypothesis that ch(heads)=r.

h₁ is the proposition that heads comes up on the first toss;

h₂ is the proposition that heads comes up on the second toss (etc.)

t₁ is the proposition that tails comes up on the first toss;

t₂ is the proposition that tails  comes up on the second toss (etc.)

t_i is equivalent to the negation of h_i:  ¬h_i.

Suppose that if the chance of the coin coming up heads/tails does not change from toss to toss. Since the tosses are independent, they obey the multiplicative rule:

(A)  Suppose that ch(heads) = 1/4. Evaluate:

(i)         ch(h₁∧h₂ )

(ii)        ch(h₁∧h₂∧h₃ )

(iii)       ch(t₁∧t₂ )

(iv)       ch(t₁∧t₂∧h₃)

Repeat (i)-(iv) on the assumption that ch(heads) = 3/4

(B)       Use the Principal Principle to calculate the following:

(i)         cr(h₁∧h₂| C^1/4)

(ii)        cr(h₁∧h₂ ∧h₃| C^1/4)

(ii)        cr(t₁∧t₂ | C^1/4)

(iv)       cr(t₁∧t₂∧h₃| C^1/4)

(v)        cr(h₁∧h₂| C^3/4)

(vi)       cr(h₁∧h₂ ∧h₃| C^3/4)

(vii)      cr(t₁∧t₂ | C^3/4)

(viii)     cr(t₁∧t₂∧h₃| C^3/4)

Suppose your prior credences in the chance hypotheses are:

(C)       What are your credences in the chance hypotheses after observing:

(i)         h₁∧h₂

(ii)        t₁∧t₂

(D)       Use the law of total probability,  the above priors, the likelihoods given by the Principal Principle, and the Law of Total Probability to evaluate the following:

(i)         cr(h₁∧h₂)

(ii)        cr(h₁∧h₂∧h₃)

(iii)       cr(t₁∧t₂)

(iv)       cr(t₁∧t₂∧h₃)

(E)       Use the definition of conditional probability and the above results to evaluate:

(i)         cr(h₃|h₁∧h₂)

(ii)        cr(h₃|t₁∧t₂)