1. clark has 5 times as many pennies as dimes. If he has a total of 4.29, which two linear equations best model this system?

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

3 years ago

Help me plz 10 points. its only middle school work. one question.

AnnZ [28]

Answer:

98

Step-by-step explanation:

6 0

3 years ago

Teresa bought 16 roses for $20.64. How much did she pay for each rose

jeka94

Roses: 16
Total Cost: $20.64

Divide:
$20.64 ÷ 16
= $1.29

Each Rose cost $1.29

Hope This Helped! :3

6 0

4 years ago

Read 2 more answers

A contractor purchases 4 dozen pairs of padded work gloves for $55.20. She vincorrectly calculates the unit price as $13.80 pe

saveliy_v [14]

Use photo math the app

6 0

3 years ago

Is this the correct answer?

andreev551 [17]

Answer:

y = 3x + 5

General Formulas and Concepts:

<u>Algebra I</u>

Slope-Intercept Form: y = mx + b

m - slope

b - y-intercept

Step-by-step explanation:

<u>Step 1: Define variables</u>

Slope <em>m</em> = 3

y-intercept <em>b</em> = 5

<u>Step 2: Write linear equation</u>

y = 3x + 5

6 0

3 years ago

1. clark has 5 times as many￼ pennies as dimes. If he has a total of 4.29, which two linear equations best model this system?

1. clark has 5 times as many pennies as dimes. If he has a total of 4.29, which two linear equations best model this system?