# Probability question?

##
A sample space #S# consists of the events #E_1, E_2, E_3, E_4, E_5.# We know #"P"(E_1)="P"(E_2)=0.3," P"(E_3)=0.1,# and #E_4# is twice as likely as #E_5# .

(a) What is the probability of event #E_4# ? of #E_5# ?

(b) Let #A= {E_1, E_3, E_4}# and #B= {E_2, E_3}.# What are #"P"(A)# and #"P"(B)# ?

(c) What is #AuuB# ?

(d) What is #AnnB# ?

A sample space

(a) What is the probability of event

(b) Let

(c) What is

(d) What is

##### 1 Answer

a)

b)

c)

d)

#### Explanation:

We assume the events

**a)** We know that the sum of all 5 single probabilities must be 1:

#color(red)([1])" " P(E_1)+P(E_2)+P(E_3)+P(E_4)+P(E_5)=1#

Given

#P(E_1)=P(E_2)=0.3#

#P(E_3)=0.1#

#P(E_4)=2P(E_5)#

we substitute these into

#0.3+0.3+0.1+2P(E_5)+P(E_5)=1#

which simplifies to

#0.7+3P(E_5)=1#

#" "3P(E_5)=0.3#

#" "P(E_5)=0.1#

And since

**b)** If

#P(A)=P(E_1 uu E_3 uu E_4)#

and since

#P(A)=P(E_1)+P(E_3)+P(E_4)#

#color(white)(P(A))=0.3+0.1+0.2#

#color(white)(P(A))=0.6#

Similarly,

**c)** We are asked to describe the **union** of events *at least* one of

#AuuB={E_1,E_2,E_3,E_4}# .

**d)** Similarly, here we are asked to find the **intersection** of *both*

#A nn B = {E_3}# .

## Note:

It is impossible for any event to have negative probability. By definition, a probability must be a number between 0 and 1 (inclusive). For example, it would be ridiculous if your local weather forecast said there was a