Question

Of all the Harry Potter books purchased in a recent year, about 60% were purchased for readers 14
or older. If 12 Harry Potter fans who bought books that year are surveyed, find the following
probabilities
At least five of them are 14 or older.
Exactly nine of them are 14 or older
Less than three of them are 14 or older.

Answer 1

Ideas?
It amounts to saying that each unspecified reader has a 60% (or 3 out of 5) chance of being 14 or older, right?

Answer 2

yeah

Answer 3

so do i take 60% of 12?

Answer 4

how would i solve the other two problems

Answer 5

Hold on there XD

It's not that simple...

Answer 6

could you explain then? Beacuse I really don't get this question

Answer 7

because

Answer 8

Independent events, heard of them?

Answer 9

yeah

Answer 10

what are they?

Answer 11

the occurrence of one event doesn't effect the chance that the other will occur or not

Answer 12

Good.
So, the status of one of the twelve readers being a 14-and-above doesn't affect the others' right?

Answer 13

yeah

Answer 14

One thing about independent events, if you want all of them to happen at the same time, you just multiply probabilities :)

Now, the probability for ONE of the readers to be a 14-and-older is 60% or 0.6, yes?

Answer 15

yes

Answer 16

The probability for at least two of them to be 14-and-older, being independent, you simply multiply the two... 0.6*0.6 = 0.36 or 36%

But you need at least FIVE.

Work it out :)

Answer 17

7.76%

Answer 18

sorry 7.78%

Answer 19

all right, hang on...

Answer 20

Wait, I think I screwed up...

Answer 21

oh ok

Answer 22

Is this binomial distribution?

Answer 23

i think

Answer 24

Even I oversimplified it LOL
Okay, clean slate

Answer 25

ok

Answer 26

Formula for binomial distribution, do you know it?

Answer 27

yeah

Answer 28

State it.

Answer 29

Okay, let's let p be the probability of the reader being 14 or older.

p = 0.6, right?

Answer 30

yeah

Answer 31

Now, the probability of there being EXACTLY r readers in the 12 being 14 or older is given by
\[\Large \left(\begin{matrix}12\\r\end{matrix} \right)p^r(1-p)^{12-r}\]

Answer 32

With this, you can answer the third question immediately.

What's the answer to the third?

Answer 33

Sorry, I meant the second one...

Answer 34

0.142

Answer 35

Good.
So much for the second.

For the first one, we'll have to add.

So, when you want the probability for 'at least 5', you might have to add "exactly 5" "exactly 6" ... and so on, until "exactly 12"

Or...

Answer 36

There is a slightly faster way :)

Answer 37

Complements.

Do you know them?

Answer 38

no. so for the first one i just have to add 0.6 five times?

Answer 39

No. You do something similar to what you did in the second question (replace r with 9), but you'll do it like...eight times (you'll have to find the probabilities from 5 to 12)

Unless... you use complements.

Answer 40

so i have to do that same equation 8 times....from 9, 8, 7, 6, etc for the values of r and then add them?

Answer 41

What you do is instead, count the instances of the event NOT happening.

So, to negate "at least 5" we need "less than 5"

So check the probability of...
-no reader 14 and above (r = 0)
-one reader (r = 1)
-two readers (r = 2)
-three readers (r= 3)
-four readers (r = 4)

And then add them up and tell me what you get.

Answer 42

so for the third one do i just do the same thing as the first and then add them up?

Answer 43

First one's not done yet. After you add up (this is the probability of the desired event NOT happening, remember?)

Subtract it from 1, and that'll be your answer to the first.

for the third, add up the probabilities involving things less than 3 (r = 0,1,2)

Answer 44

and then i will be done?