Question

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`?

Answer 1

a) `P(E_4)=0.2; " "P(E_5)=0.1.`
b) `P(A)=0.6;" " P(B)=0.4.`
c) `E_1, E_2, E_3, E_4`
d) `E_3`

Explanation:

We assume the events `E_1` through `E_5` are mutually exclusive and cover all of `S`. (Like how when you roll a die, you must land on one and only one number.)

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 `color(red)([1])`:

`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 `P(E_4)=2P(E_5)`, we get `P(E_4)=2(0.1)=0.2.`

b) If `A={E_1, E_3, E_4},` then

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

and since `E_1,E_3,` and `E_4` are mutually exclusive, we get

`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, `P(B)=P(E_2)+P(E_3)=0.4.`

c) We are asked to describe the union of events `A` and `B.` This consists of all the events that show up in at least one of `A` or `B` (and potentially both). Observing the elements of both `A` and `B`, we find

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

d) Similarly, here we are asked to find the intersection of `A` and `B,` which consists of all the events that appear in both `A` and `B.` Since the only event to appear in both `A` and `B` is `E_3,` we can say

`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 `–15%` chance of rain today.