A UMBC student has two different email accounts. Suppose that 70% of her messages come into Account 1. Of the messages that come into Account 1.5% are spam. Meanwhile, of the messages that come into Account 2.8% are spam.

a) If a message is randomly selected, what is the probability that it is NOT a spam?
b) If a randomly selected message is a spam, what is the probability that it came from Account 1.
c) Suppose 5 messages are randomly selected. What is the probability that exactly 2 of them are spam? dy
d) Suppose 20 messages are randomly selected. What is the probability that at least 3 of them are not spam?

a) 0.941

b) 0.593

c) 0.02901

d) 0.9999999999

Step-by-step explanation:

A student has two accounts.

70% of her messages come from account 1.

Probability that a message is on account 1 = P(A) = 0.7

Meaning 30% of her messages come from account 2.

Probability that a message is on account 2 is P(B) = 0.3

Of the messages that come into Account 1, 5% are spam

Calculating in terms of total percentage of messages.

The amount of spam on account 1 is 5% of 70% of the total amount of messages.

P(A n S) = 0.05 × 0.7 = 0.035 = 3.5%

The amount of spam on account 2 is 8% of 70% of the total amount of messages.

P(B n S) = 0.08 × 0.3 = 0.024 = 2.4%

Total amount of spam messages received = P(S) = 2.4% + 3.5% = 5.9% = 0.059

a) If a message is randomly selected, what is the probability that it is NOT a spam?

P(S') = 1 - P(S) = 1 - 0.059 = 0.941

b) If a randomly selected message is a spam, what is the probability that it came from Account 1.

P(A|S) = P(A n S)/P(S)

P(A|S) = (0.035/0.059)

P(A|S) = 0.593

c) Suppose 5 messages are randomly selected. What is the probability that exactly 2 of them are spam

This is a binomial distribution problem

Binomial distribution function is represented by

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = 5 messages picked

x = Number of successes required = number of spam messages that should be amongst the 5 messages picked = 2

p = probability of success = probability that a message is a spam message = 0.059

q = probability of failure = probability that a message is NOT a spam message = 0.941

P(X=2) = ⁵C₂ (0.059)² (0.941)⁵⁻²

P(X=2) = 0.02901

d) Suppose 20 messages are randomly selected. What is the probability that at least 3 of them are not spam?

This is also a binomial distribution problem

Binomial distribution function is represented by

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = number of messages = 20

x = Number of successes required = at least 3

p = probability of success = probability that a message is not a spam = 0.941

q = probability of failure = probability that a message is a spam message = 0.059

P(X ≥ 3) = 1 - P(X < 3)

P(X < 3) = P(X=0) + P(X=1) + P(X=2) = 0.0000000000001

P(X ≥ 3) = 1 - P(X < 3) = 1 - 0.0000000000001 = 0.9999999999 (Almost 1!)

Hope this Helps!!!

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: