Question

Suppose you get 7 heads in 8 flips. This seems odd... I mean on average you expect
to get half heads and half tails. But then again odd things happen. Certainly in
the history of the humankind, a fair coin has been flipped 8 times with a result of
7 heads. Before accusing your friend of being a cheater, let’s find the probability of
something so odd happening.
i. If the the coin is fair, what is the probability of getting a result so far (or farther)
from the “expected” result of 4 heads?

Answer 1

Is it the same as saying "What is the probability of getting 5 or MORE heads? But I wasnt sure on how to go solve that...

Answer 2

any lucky guys?

Answer 3

You simply need to find the probability of getting EXACTLY 7 heads

Answer 4

Since we're doing the experiment 8 times,
consider an 8 letter string

Answer 5

Hmm...but its asking me what the probability is of getting anything that is farther than 4 (the average number of heads)

Answer 6

There was a remark the teacher added here after the question that said.. "Remark: The distinction between computing “the probability of getting a result
exactly as far” and “the probability of getting a result so far (or farther)”
is important. When flipping 20 coins, for example, the probability of getting
exactly 12 heads is only about 12%, but 12 heads is not a very unusual result
when flipping a coin 20 times – in fact when flipping a coin 20 times you’ll getting
12 or more heads about 25% of the time and 8 or fewer heads about 25% of the
time, meaning you get a result as far or farther (as 12 is) from the expected value
of 10 about 50% of the time."

Answer 7

So you want to find the probability that the number of heads is greater than 4 ?

Answer 8

Consider an 8 letter string :

x x x x x x x x

Answer 9

Each letter has two options : H or T

Total how many different strings are possible ?

Answer 10

256

Answer 11

How did you get 256 ?

Answer 12

That is the possible outcome when we flip a coin 8 times. 2^k = 2^8 = 256 since each time there is a possibility of two outcomes

Answer 13

Good. Next look at how many of them have exactly 5 H

Answer 14

If a string has exactly 5 H,
it must have exactly 3 H

Answer 15

True....But thats where I am stuck.

Answer 16

5 heads to fill. And you have 8 empty spaces to choose from

Answer 17

How many ways can you choose 5 spaces from the available 8 spaces ?

Answer 18

How would I solve that..? LIke there are too many possibilities for me to write it down?

Answer 19

Lets go through a simpler example first

Answer 20

Imagine you went to a restaurant to have breakfast

Answer 21

You're too much hungry that you want to eat 2 items

Answer 22

The menu has 3 different items.

How many different combinations are possible ?

Answer 23

3^3? = 27

Answer 24

Nope. Forget math. Think like a illeterate man/woman

Answer 25

You have 3 different items to choose from.

You want to pick 2 items.

How many different combinations are possible ?

Answer 26

Maybe label the items as : A, B, C

Answer 27

Which two of the above letters you like most ?

Answer 28

We can make combinations AB, AC, BC, BA, CA and CB?

Answer 29

Good. Here the order doesn't matter. AB is same as BA. End of the day, you will remember what things you ate for breakfast. Not the order in which you have eaten them.

Answer 30

Since the order doesn't matter, how many combinations are possible ?

Answer 31

AB, CA, AND CB

Answer 32

Excellent! so there are exactly 3 ways to choose 2 items from 3 items

Answer 33

Can we generalize this

Answer 34

How many ways can you pick "r" items from a collection of "n" items ?

Answer 35

Are you asking for a equation..?

Answer 36

wow...that was lucky

Answer 37

Yep! He is asking for an equation to generalise the example given by him.

Answer 38

Have you studied the combination stuff??

Answer 39

Nope...Im pretty lost on probability.

Answer 40

Ok! You might have found that there were three cases possible for you to have your breakfast. Right?
So that simply means 3C2 combinations were possible.
So far clear to you??

Answer 41

Yes :) So...Would it be 8C5?

Answer 42

Great!

Answer 43

So would I do...
"8C5 + 8C6 + 8C7 + 8C8" to get the probability of five or more?

Answer 44

Correct!
Remember these are just the possible outcomes. To evaluate the probability you would have to divide them by total cases.

Answer 45

Wow. YOU ARE A LIFESAVER.

Answer 46

Thank you so muchhhh! Daniyavad!

Answer 47

Welcum;)

Answer 48

Awesome !

Answer 49

One last question guys.

Answer 50

Since its asking what the probability I wont get 4 or heads is...Would I also have to add "8C1 + 8C2 +8C3" to 8C5+..8C8"

Answer 51

It was your question. You should ask your teacher

Answer 52

Also "8C5 + 8C6 + 8C7 + 8C8" is not the probability

Answer 53

Remember, probability is always between 0 and 1.

It is NEVER greater than 1.

Answer 54

Right! I will have to divide it by 256

Answer 55

good

Answer 56

Thank you so much guys. Wish I could give both of you medals! :)

Answer 57

Give it to him!