ae:T53a,

P(F) \ne 0

72:["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$ae"}}] 73:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["What does \"given ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

F

"}}],"\" do to the sample space?"]}]]}] 74:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Shrinks it to ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

F

"}}],"; outcomes in ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

F

"}}]," stay equally likely."]}]]}] 75:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

P(F \\mid F)

"}}],"?"]}]]}] 76:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

1

"}}],"."]}]]}] af:T5c1,

P(E' \mid F)

77:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$af"}}]," in terms of ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

P(E \\mid F)

"}}],"?"]}]]}] b0:T51b,

1 - P(E \mid F)

78:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b0"}}],"."]}]]}] b1:T71e,

P(A \mid B) = P(B \mid A)

79:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Is ",["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b1"}}]," in general?"]}]]}] 7a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"No: same numerator, different denominators."}]]}] 7b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Two-way table, \"given cricket\": divide by?"}]]}] 7c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"The cricket row (or column) total, not the grand total."}]]}] 7d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Die shows an even number. ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

P(2)

"}}],"?"]}]]}] b2:T4f9,

\dfrac{1}{3}

7e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b2"}}],"."]}]]}] 7f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Two dice, at least one six: how many outcomes?"}]]}] 80:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

11

"}}]," (not ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

12

"}}],": double six counted once)."]}]]}] 81:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Multiplication rule for two events?"}]]}] b3:T8bc,

P(E \cap F) = P(E)\,P(F \mid E)

82:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b3"}}],"."]}]]}] 83:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Multiplication rule for three events?"}]]}] b4:T931,

P(E)\,P(F \mid E)\,P(G \mid E \cap F)

84:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b4"}}],"."]}]]}] 85:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Independent events: definition?"}]]}] b5:T75b,

P(E \cap F) = P(E)\,P(F)

86:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b5"}}],"."]}]]}] 87:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Mutually exclusive events: definition?"}]]}] b6:T4fe,

P(E \cap F) = 0

88:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b6"}}],": they cannot happen together."]}]]}] 89:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Can events with non-zero probabilities be both mutually exclusive and independent?"}]]}] 8a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"No."}]]}] 8b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["If ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

E

"}}],", ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

F

"}}]," are independent, are ",["$","span","code-2",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

E'

"}}]," and ",["$","span","code-3",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

F'

"}}],"?"]}]]}] 8c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Yes."}]]}] 8d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"\"At least one\" for independent events?"}]]}] b7:T943,

1 - P(E')\,P(F')\cdots

8e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b7"}}]}]]}] 8f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Without replacement: what changes in the second factor?"}]]}] 90:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"The total (and possibly the favourable count) drops by one."}]]}] 91:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"King and heart from one card: independent?"}]]}] b8:Te30,

\dfrac{1}{52} = \dfrac{1}{13} \times \dfrac{1}{4}

92:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Yes: ",["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b8"}}],"."]}]]}] 93:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Picture: a tree diagram. How do you get a path probability?"}]]}] 94:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Multiply the branch probabilities along the path."}]]}] 95:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Mutual independence of three events needs?"}]]}] b9:T92e,

P(E \cap F \cap G) = P(E)P(F)P(G)

96:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["All three pairs independent ",["$","strong","strong-0",{"children":"and"}]," ",["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b9"}}],"."]}]]}] 97:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Partition of ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

S

"}}],"?"]}]]}] 98:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Disjoint events with non-zero probabilities whose union is ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

S

"}}],"."]}]]}] 99:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Theorem of total probability?"}]]}] ba:Tc0d,

P(A) = \sum P(E_i)\,P(A \mid E_i)

9a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$ba"}}],"."]}]]}] 9b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Bayes' theorem?"}]]}] bb:T1c0a,

P(E_i \mid A) = \dfrac{P(E_i)P(A \mid E_i)}{\sum_j P(E_j)P(A \mid E_j)}

9c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$bb"}}]}]]}] 9d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Prior, likelihood, posterior?"}]]}] bc:T4a8,

P(E_i)

bd:T5ea,

P(A \mid E_i)

be:T5ea,

P(E_i \mid A)

9e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$bc"}}],", ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$bd"}}],", ",["$","span","code-2",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$be"}}],"."]}]]}] 9f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Posteriors over a partition add to?"}]]}] a0:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

1

"}}],"."]}]]}] a1:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Picture: a partition with an event ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

A

"}}]," crossing the strips. What does it show?"]}]]}] bf:T51c,

A \cap E_i

a2:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

A

"}}]," splits into disjoint pieces ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$bf"}}],": total probability."]}]]}] a3:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Bags I (",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

5W

"}}],", ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

3B

"}}],") and II (",["$","span","code-2",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

2W

"}}],", ",["$","span","code-3",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

6B

"}}],"), coin choice. ",["$","span","code-4",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

P(W)

"}}],"?"]}]]}] c0:T4fc,

\dfrac{7}{16}

c1:T877,

P(\text{I} \mid W) = \dfrac{5}{7}

a4:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$c0"}}],"; and ",["$","span","code-1",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$c1"}}],"."]}]]}] a5:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Rare condition, good test: is a positive result conclusive?"}]]}] a6:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"No: false alarms from the large healthy group can outnumber true positives."}]]}] a7:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Bayes long answer: the five marks?"}]]}] a8:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Events, priors and likelihoods, total, ratio, conclusion."}]]}] a9:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Total probability or Bayes? \"Find the probability that the ball is white.\""}]]}] aa:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Total probability (effect through several causes)."}]]}] ab:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Which topics of this chapter are deleted for 2026–27?"}]]}] ac:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Random variables, probability distributions, mean and variance, Bernoulli trials and the binomial distribution."}]]}]

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