Grade 10 Probability Venn Diagram Question

Andy Soper

October 15, 2013

This document was constructed and type-set using PCTEX(a dielect of LATEX)

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1 Grade 10 Probability

1.1 Venn Diagram / Boolean Algebra Question

Figure 1: Venn Diagram and Boolean Algebra

(a) Complementary eventa are mutually exclusive (if one happens the othercannot happen) and exhaustive (the events cover the entire Sample Space (oneor the other must occur. There is no other possibility.)

The two events are compliments if

1. the sum of their probabilities equals 1

2. they are disjoint (no overlap)

3. the combined probabilities of the rtwo events equals 1

4. they are independent events and their probabilities are equal.

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Hence ...

1. P (A) + P (B) = 1

2. A ∩B = The empty set. Therefore A’ = B, B’ = A

3. nA+nBnS = 1

4. P (A) = P (B)

——————————————————————

(a) nA + nB = nS = 80; nA = nB = 40; x+15 = 40; x=25

(b) P(A) = 25% = 0,25; x+1580 = 0, 25; x+ 15 = 20; x=5 =⇒

(c) P (A′ ∩B′) = P (A′)× P (B′) = 15

80−x−1580 × 80−60

80 = 15 ; (65− x)× 20 = 6400

5 = 1280

1300− 20x = 1280; −20x = −20x = 1 =⇒ x=1 satisfies row 2 of (c)

(d) P (B′ ∩A) = P (A∪B)′ ⇒ P (B′)×P (A) = 1− [P (A)+P (B)−P (A)×P (B)]

80−6080 × x+15

80 = 1− [x+1580 + 60

80 −x+1580 × 60

80 ]

2080 ×

x+1580 = 1− [x+75

80 − x+1580 × 60

80 ]

2080 ×

x+1580 + [x+75

80 − x+1580 × 60

80 ] = 1

Multiply all terms by 80× 80

20(x + 15) + 80(x + 75)− 60(x + 15) = 6400

20x + 300 + 80x + 6000− 60x− 900 = 6400

40x + 5400 = 6400

40x = 1000 x = 25 =⇒ x=25 satisfies the third row of (d)

Lesson Learned: P (A′) = nS−nAnS

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Figure 2: Question2

Figure 3: Question 2 Venn Diagram

(a) n(A’) = 60 - 20 = 40 (b) n(B’) = 60 - 30 = 30 (c) n(A and B’) = 10

(d) n(A’ and B) = 20 (e) n(A or B) = 40 as they define union

(f) n(A or B’) = 20 + 20 = 40

In Progress ...

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