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1 year ago

Which point represents - 12 on the number line?

Mathematics

1 answer:

faust18 [17]1 year ago

6 0

Answer:

• Draw a number line.

•• Spot the number -12 on the number line.

••• Mark your point at -12.

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For two events A and B show that P (A∩B) ≥ P (A)+P (B)−1.

nordsb [41]

Answer:

<h3>For two events A and B show that P (A∩B) ≥ P (A)+P (B)−1.</h3>

By De morgan's law

$(A\cap B)^{c}=A^{c}\cup B^{c}\\\\P((A\cap B)^{c})=P(A^{c}\cup B^{c})\leq P(A^{c})+P(B^{c}) \\\\1-P(A\cap B)\leq P(A^{c})+P(B^{c}) \\\\1-P(A\cap B)\leq 1-P(A)+1-P(B)\\\\-P(A\cap B)\leq 1-P(A)-P(B)\\\\P(A\cap B)\geq P(A)+P(B)-1$

which is Bonferroni’s inequality

<h3>Result 1: P (Ac) = 1 − P(A)</h3>

Proof

If S is universal set then

$A\cup A^{c}=S\\\\P(A\cup A^{c})=P(S)\\\\P(A)+P(A^{c})=1\\\\P(A^{c})=1-P(A)$

<h3>Result 2 : For any two events A and B, P (A∪B) = P (A)+P (B)−P (A∩B) and P(A) ≥ P(B)</h3>

Proof:

If S is a universal set then:

$A\cup(B\cap A^{c})=(A\cup B) \cap (A\cup A^{c})\\=(A\cup B) \cap S\\A\cup(B\cap A^{c})=(A\cup B)$

Which show A∪B can be expressed as union of two disjoint sets.

If A and (B∩Ac) are two disjoint sets then

$P(A\cup B) =P(A) + P(B\cap A^{c})---(1)\\$

B can be expressed as:

$B=B\cap(A\cup A^{c})\\$

If B is intersection of two disjoint sets then

$P(B)=P(B\cap(A)+P(B\cup A^{c})\\P(B\cup A^{c}=P(B)-P(B\cap A)$

Then (1) becomes

$P(A\cup B) =P(A) +P(B)-P(A\cap B)\\$

<h3>Result 3: For any two events A and B, P(A) = P(A ∩ B) + P (A ∩ Bc)</h3>

Proof:

If A and B are two disjoint sets then

$A=A\cap(B\cup B^{c})\\A=(A\cap B) \cup (A\cap B^{c})\\P(A)=P(A\cap B) + P(A\cap B^{c})\\$

<h3>Result 4: If B ⊂ A, then A∩B = B. Therefore P (A)−P (B) = P (A ∩ Bc) </h3>

Proof:

If B is subset of A then all elements of B lie in A so A ∩ B =B

$A =(A \cap B)\cup (A\cap B^{c}) = B \cup ( A\cap B^{c})$

where A and A ∩ Bc are disjoint.

$P(A)=P(B\cup ( A\cap B^{c}))\\\\P(A)=P(B)+P( A\cap B^{c})$

From axiom P(E)≥0

$P( A\cap B^{c})\geq 0\\\\P(A)-P(B)=P( A\cap B^{c})\\P(A)=P(B)+P(A\cap B^{c})\geq P(B)$

Therefore,

P(A)≥P(B)

8 0

2 years ago

The population in the current year is 31.5 million and the real GDP is $814 million. The previous year's statistics were a popul

andrew11 [14]

Answer:

The change in the standard of living, measured by growth in real GDP per person, is 0.1356%

Step-by-step explanation:

Given,

In the current year,

Population = 31.5 million,

Real GDP = $814 million,

So, the real GDP per person = $\frac{814}{31.5}$ ≈ 25.841

In the previous year,

Population = 31 million,

Real GDP = $800 million,

So, the real GDP per person = $\frac{800}{31}$ ≈ 25.806

Hence, change percentage = $\frac{\text{Change in GDP per person}}{\text{Previous year GDP per person}}\times 100$

$=\frac{25.841-25.806}{25.806}\times 100$

$=\frac{0.035}{25.806}\times 100$

$=\frac{3.5}{25.806}$

≈ 0.1356 %

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2 years ago

What is 9 3/7 in simplest form?

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Answer:

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Step-by-step explanation:

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If AB = CD, then CD = AB. symmetric property

If MN = XY , and XY = AB , then MN = AB. transitive property

Segment CD is congruent to segment CD reflexive property

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3 years ago

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Which point represents - 12 on the number line? ​

Which point represents - 12 on the number line?