Question

You're playing a game where you defend your village for me or evasion there are three characters of habit or human and five defensive tools magic sword shield slingshot or umbrella to pick from if you randomly choose your character into what is your probability that you won't be a hobby or use an umbrella

Answer 1

`P(Not\ hobbit\ or\ Not\ Umbrella) = (14)/(15)`

Explanation:

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Given

`x = \{elf, hobbit, human\}` ---- characters

`n(x) =3`

`y = \{Magic,Sword,Shield,Slingshot,Umbrella\}` --- tools

`n(y) = 5`

Required

The probability that your character won't be a hobbit or your tool won't be an umbrella?

First we calculate the probability of choosing a hobbit.

`x = \{elf, hobbit, human\}`

`n(hobbit) = 1`

So:

`P(hobit) = (n(hobbit))/(n(x))`

`P(hobit) = (1)/(3)`

Using the complement rule, we calculate the probability of not choosing a hobbit

`P(Not\ hobbit) = 1 - P(hobbit)`

`P(Not\ hobbit) = 1 - (1)/(3)`

`P(Not\ hobbit) =(2)/(3)`

Next, we calculate the probability of choosing an umbrella.

`y = \{Magic,Sword,Shield,Slingshot,Umbrella\}`

`n(Umbrella) = 1`

So:

`P(Umbrella) = (n(Umbrella))/(n(y))`

`P(Umbrella) = (1)/(5)`

Using the complement rule, we calculate the probability of not choosing an umbrella

`P(Not\ Umbrella) = 1 - \P(Umbrella)`

`P(Not\ Umbrella) = 1 - (1)/(5)`

`P(Not\ Umbrella) = (4)/(5)`

The required probability is calculated using:

`P(A\ or\ B) = P(A) + P(B) - P(A)P(B)`

In this case;

`P(Not\ hobbit\ or\ Not\ Umbrella) = P(Not\ hobbit) + P(Not\ Umbrella) -`
`P(Not\ hobbit)*P(Not\ Umbrella)`

`P(Not\ hobbit\ or\ Not\ Umbrella) = (2)/(3) + (4)/(5) - (2)/(3) * (4)/(5)`

`P(Not\ hobbit\ or\ Not\ Umbrella) = (2)/(3) + (4)/(5) - (8)/(15)`

Take LCM

`P(Not\ hobbit\ or\ Not\ Umbrella) = (10+12-8)/(15)`

`P(Not\ hobbit\ or\ Not\ Umbrella) = (14)/(15)`