Witch Expression gives the closest estimate of 228-179

An e-mail filter is planned to separate valid e-mails from spam. The word free occurs in 60% of the spam messages and only 4% of

ANEK [815]

Answer:

(a) 0.152

(b) 0.789

(c) 0.906

Step-by-step explanation:

Let's denote the events as follows:

F = The word free occurs in an email

S = The email is spam

V = The email is valid.

The information provided to us are:

The probability of the word free occurring in a spam message is, $P(F|S)=0.60$
The probability of the word free occurring in a valid message is, $P(F|V)=0.04$
The probability of spam messages is,

$P(S)=0.20$

First let's compute the probability of valid messages:

$P (V) = 1 - P(S)\\=1-0.20\\=0.80$

(a)

To compute the probability of messages that contains the word free use the rule of total probability.

The rule of total probability is:

$P(A)=P(A|B)P(B)+P(A|B^{c})P(B^{c})$

The probability that a message contains the word free is:

$P(F)=P(F|S)P(S)+P(F|V)P(V)\\=(0.60*0.20)+(0.04*0.80)\\=0.152\\$

The probability of a message containing the word free is 0.152

(b)

To compute the probability of messages that are spam given that they contain the word free use the Bayes' Theorem.

The Bayes' theorem is used to determine the probability of an event based on the fact that another event has already occurred. That is,

$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$

The probability that a message is spam provided that it contains free is:

$P(S|F)=\frac{P(F|S)P(S)}{P(F)}\\=\frac{0.60*0.20}{0.152} \\=0.78947\\$

The probability that a message is spam provided that it contains free is approximately 0.789.

(c)

To compute the probability of messages that are valid given that they do not contain the word free use the Bayes' Theorem. That is,

$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$

The probability that a message is valid provided that it does not contain free is:

$P(V|F^{c})=\frac{P(F^{c}|V)P(V)}{P(F^{c})} \\=\frac{(1-P(F|V))P(V)}{1-P(F)}\\=\frac{(1-0.04)*0.80}{1-0.152} \\=0.90566$

The probability that a message is valid provided that it does not contain free is approximately 0.906.

4 0

3 years ago

state whether the slope of a line passing through (2, -3) and (1, 8) is positive, negative, zero or undefined.

Natalija [7]

Slope = change in y / change in x

= (8 - -3) / (1 - 2) = (8 + 3) / ( 1 - 2)

= 11 / -1

= -11

Slope is negative.

4 0

3 years ago

Please Hurry) Consider the quadratic equation 3x^2 - 6 = 2x. A) What is the value of the discrimination? B) What does the discri

Levart [38]

Answer:

A. 76

B. We will get two real solutions for the equation.

Step-by-step explanation:

A

For any quadratic equation ax²+bx+c=0, the discriminant is b²-4ac.

In 3x²-2x-6=0, a=3, b=-2 and c=-6:

b²-4ac

(-2)²-4(3)(-6)=4+72=76

B.

When the discriminate is positive we get two real solutions to the equation, when it's zero we get one real solution, and when it's negative we get two complex solutions. Since our discriminate, 76, is positive, we know that we will get two real solutions to 3x²-6=2x.

3 0

3 years ago

Simplify : (√x + √y ) (√x − √y) (x + y)(x2 + y2)

Ivan

$Solution, \mathrm{Expand}\::\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)\left(x+y\right)\left(x^2+y^2\right):\quad :x^4-:y^4$

$Steps:$

$\mathrm{Expand}\:\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right):\quad x-y, =\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)$

$\mathrm{Expand}\:\left(x-y\right)\left(x+y\right):\quad x^2-y^2, =:\left(x^2-y^2\right)\left(x^2+y^2\right)$

$\mathrm{Expand}\:\left(x^2-y^2\right)\left(x^2+y^2\right):\quad x^4-y^4, =:\left(x^4-y^4\right)$

$\mathrm{Expand}\::\left(x^4-y^4\right):\quad :x^4-:y^4, =:x^4-:y^4$

The correct answer is $x^4-y^4$

Hope this helps!!!

4 0

3 years ago

Madelyn invested $650 in an account paying an interest rate of 5 3/4 % compounded daily. Jonathan invested $650 in an account pa

Olegator [25]

Answer: Madelyn would have $14.17 in her account more than Jonathan.

Step-by-step explanation:

We would apply the formula for determining compound interest which is expressed as

A = P(1+r/n)^nt

Where

A = total amount in the account at the end of t years

r represents the interest rate.

n represents the periodic interval at which it was compounded.

P represents the principal or initial amount deposited

Considering Madelyn's investment,

P = 650

r = 5.75% = 5.75/100 = 0.0575

n = 365 because it was compounded 12 times in a year.

t = 10 years

Therefore,

A = 650(1+0.0575/365)^365 × 10

A = 650(1+0.0001575)^3650

A = 650(1.0001575)^3650

A = 650 × 1.7768

A = 1154.92

The formula for continuously compounded interest is

A = P x e (r x t)

Where

A represents the future value of the investment after t years.

P represents the present value or initial amount invested

r represents the interest rate

t represents the time in years for which the investment was made.

e is the mathematical constant approximated as 2.7183.

Considering Jonathan's investment,,

P = 650

r = 5.625% = 5.625/100 = 0.05625

t = 10 years

Therefore,

A = 650 x 2.7183^(0.05625 x 10)

A = 650 x 2.7183^(0.5625)

A = 650 × 1.755

A = $1140.75

The difference in the amount in both accounts is

1154.92 - 1140.75 = $14.17

5 0

3 years ago