# data[].attributes.body
Watch this video for an introduction to the concept of Arithmetic Sequences

# data[].attributes.attachment_link.title
Intro to arithmetic sequences | Algebra (video) | Khan Academy

# data[].attributes.attachment_link.description
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

# data[].attributes.body
This website discusses compound events in depth and also provides great examples of other probability concepts.

# data[].attributes.attachment_link.description
A compound event is an event that includes two or more simple events. Simple events are events that can have only one outcome, while compound events can have multiple different outcomes. Compound events can be made up of a number of independent events (events in which the outcome of one event has no…

# data[].attributes.body
Check out the overview of Independent Events in Statistics. *Special note - This resource also discusses Dependent Events.*

# data[].attributes.attachment_link.readable_text
Life is full of random events!

You need to get a "feel" for them to be a smart and successful person.

The toss of a coin, throwing dice and lottery draws are all examples of random events.

There can be:

Dependent Events where what happens depends on what happened before, such as taking cards from a deck makes less cards each time (learn more at Conditional Probability), or

Independent Events which we learn about here.

Independent Events

Independent Events are not affected by previous events.
This is an important idea!
A coin does not "know" it came up heads before.

And each toss of a coin is a perfect isolated thing.

Example: You toss a coin and it comes up "Heads" three times ... what is the chance that the next toss will also be a "Head"?
The chance is simply ½ (or 0.5) just like ANY toss of the coin.
What it did in the past will not affect the current toss!

Some people think "it is overdue for a Tail", but really truly the next toss of the coin is totally independent of any previous tosses.

Saying "a Tail is due", or "just one more go, my luck is due to change" is called The Gambler's Fallacy

Of course your luck may change, because each toss of the coin has an equal chance.

Probability of Independent Events

"Probability" (or "Chance") is how likely something is to happen.

So how do we calculate probability?

Probability of an event happening = Number of ways it can happen Total number of outcomes

Example: what is the probability of getting a "Head" when tossing a coin?

Number of ways it can happen: 1 (Head)

Total number of outcomes: 2 (Head and Tail)

Example: what is the probability of getting a "4" or "6" when rolling a die?

Number of ways it can happen: 2 ("4" and "6")

Total number of outcomes: 6 ("1", "2", "3", "4", "5" and "6")

Ways of Showing Probability

Probability goes from 0 (imposssible) to 1 (certain):

It is often shown as a decimal or fraction.

Example: the probability of getting a "Head" when tossing a coin:

Two or More Events

We can calculate the chances of two or more independent events by multiplying the chances.

Example: Probability of 3 Heads in a Row

For each toss of a coin a Head has a probability of 0.5:

And so the chance of getting 3 Heads in a row is 0.125

So each toss of a coin has a ½ chance of being Heads, but lots of Heads in a row is unlikely.

Example: Why is it unlikely to get, say, 7 heads in a row, when each toss of a coin has a ½ chance of being Heads?

Because we are asking two different questions:

Question 1: What is the probability of 7 heads in a row?

Question 2: When we have just got 6 heads in a row, what is the probability that the next toss is also a head?

You can have a play with the Quincunx to see how lots of independent effects can still have a pattern.

Notation

We use "P" to mean "Probability Of",

So, for Independent Events:

P(A and B) = P(A) × P(B)

Probability of A and B equals the probability of A times the probability of B

Example: your boss (to be fair) randomly assigns everyone an extra 2 hours work on weekend evenings between 4 and midnight.

What are the chances you get Saturday between 4 and 6?

Day: there are two days on the weekend, so P(Saturday) = 0.5

Time: you want the 2 hours of "4 to 6", out of the 8 hours of 4 to midnight):
P("4 to 6") = 2/8 = 0.25
And:

Or a 12.5% Chance

(Note: we could ALSO have worked out that you wanted 2 hours out of a total possible 16 hours, which is 2/16 = 0.125. Both methods work here.)

Another Example

Example: the chance of a flight being delayed is 0.2 (=20%), what are the chances of no delays on a round trip

The chance of a flight not having a delay is 1 − 0.2 = 0.8, so these are all the possible outcomes:

When we add all the possibilities we get:

0.64 + 0.16 + 0.16 + 0.04 = 1.0

They all add to 1.0, which is a good way of checking our calculations.

Result: 0.64, or a 64% chance of no delays

One More Example

Imagine there are two groups:

A member of each group gets randomly chosen for the winners circle,

then one of those gets randomly chosen to get the big money prize:

What is your chance of winnning the big prize?

there is a 1/5 chance of going to the winners circle

and a 1/2 chance of winning the big prize

So you have a 1/5 chance followed by a 1/2 chance ... which makes a 1/10 chance overall:

Or we can calculate using decimals (1/5 is 0.2, and 1/2 is 0.5):

0.2 x 0.5 = 0.1

So your chance of winning the big money is 0.1 (which is the same as 1/10).

Coincidence!

Many "Coincidences" are, in fact, likely.

Example: you are in a room with 30 people, and find that Zach and Anna celebrate their birthday on the same day.

Do you say:

"Wow, how strange !", or

"That seems reasonable, with so many people here"

In fact there is a 70% chance that would happen ... so it is likely.

Why is the chance so high?

Because you are comparing everyone to everyone else (not just one to many).

And with 30 people that is 435 comparisons

(Read Shared Birthdays to find out more.)

Example: Snap!

Did you ever say something at exactly the same time as someone else?

Wow, how amazing!

But you were probably sharing an experience (movie, journey, whatever) and so your thoughts were similar.

And there are only so many ways of saying something ...

... so it is like the card game "Snap!" (also called Slaps or Slapjack) ...

... if you speak enough words together, they will eventually match up.

So, maybe not so amazing, just simple chance at work.

Can you think of other cases where a "coincidence" was simply a likely thing?

Conclusion

# data[].attributes.attachment_link.description
Life is full of random events! You need to get a "feel" for them to be a smart and successful person. The toss of a coin, throwing dice and lottery draws are all examples of random events.

# data[].attributes.body
Use the written description and the illustrations to review the idea of the Line of Best Fit.

# data[].attributes.attachment_link.description
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

# data[].attributes.headline
Terms #5 and #6: Multiplicative Inverse and Additive Inverse

# data[].attributes.subject
Terms #5 and #6: Multiplicative Inverse and Additive Inverse

# data[].attributes.body
Check out the quick explanation of additive inverse and multiplicative inverse.

# data[].attributes.attachment_link.description
Create a new document and edit with others at the same time -- from your computer, phone or tablet. Get stuff done with or without an internet connection. Use Docs to edit Word files. Free from Google.

# data[].attributes.headline
Terms #3 and #4: Permutation and Combination

# data[].attributes.subject
Terms #3 and #4: Permutation and Combination

# data[].attributes.body
Read carefully about the differences between permuations and combinations.

# data[].attributes.attachment_link.readable_text
What's the Difference?

In English we use the word "combination" loosely, without thinking if the order of things is important. In other words:

"My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

"The combination to the safe is 472". Now we do care about the order. "724" won't work, nor will "247". It has to be exactly 4-7-2.

So, in Mathematics we use more precise language:

When the order doesn't matter, it is a Combination.

When the order does matter it is a Permutation.

So, we should really call this a "Permutation Lock"!

In other words:

A Permutation is an ordered Combination.

To help you to remember, think " Permutation ... P osition"

Permutations

There are basically two types of permutation:

1. Repetition is Allowed: such as the lock above. It could be "333".

2. No Repetition: for example the first three people in a running race. You can't be first and second.

1. Permutations with Repetition

These are the easiest to calculate.

When a thing has n different types ... we have n choices each time!

For example: choosing 3 of those things, the permutations are:

n × n × n (n multiplied 3 times)

More generally: choosing r of something that has n different types, the permutations are:
n × n × ... (r times)
(In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.)

Which is easier to write down using an exponent of r:
n × n × ... (r times) = nr
Example: in the lock above, there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 3 of them:
10 × 10 × ... (3 times) = 10³ = 1,000 permutations
So, the formula is simply:

2. Permutations without Repetition

In this case, we have to reduce the number of available choices each time.

Example: what order could 16 pool balls be in?

After choosing, say, number "14" we can't choose it again.

So, our first choice has 16 possibilites, and our next choice has 15 possibilities, then 14, 13, 12, 11, ... etc. And the total permutations are:
16 × 15 × 14 × 13 × ... = 20,922,789,888,000
But maybe we don't want to choose them all, just 3 of them, and that is then:
16 × 15 × 14 = 3,360
In other words, there are 3,360 different ways that 3 pool balls could be arranged out of 16 balls.
Without repetition our choices get reduced each time.
But how do we write that mathematically? Answer: we use the " factorial function"

The factorial function (symbol: ) just means to multiply a series of descending natural numbers. Examples:

4! = 4 × 3 × 2 × 1 = 24

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

1! = 1

Note: it is generally agreed that 0! = 1. It may seem funny that multiplying no numbers together gets us 1, but it helps simplify a lot of equations.

So, when we want to select all of the billiard balls the permutations are:
16! = 20,922,789,888,000
But when we want to select just 3 we don't want to multiply after 14. How do we do that? There is a neat trick: we divide by 13!

That was neat: the 13 × 12 × ... etc gets "cancelled out", leaving only 16 × 15 × 14.

The formula is written:

Example Our "order of 3 out of 16 pool balls example" is:

(which is just the same as: 16 × 15 × 14 = 3,360)

Example: How many ways can first and second place be awarded to 10 people?

(which is just the same as: 10 × 9 = 90)

Notation

Instead of writing the whole formula, people use different notations such as these:

Examples:

Combinations

There are also two types of combinations (remember the order does not matter now):

1. Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)

2. No Repetition: such as lottery numbers (2,14,15,27,30,33)

1. Combinations with Repetition

Actually, these are the hardest to explain, so we will come back to this later.

2. Combinations without Repetition

This is how lotteries work. The numbers are drawn one at a time, and if we have the lucky numbers (no matter what order) we win!

The easiest way to explain it is to:

assume that the order does matter (ie permutations),

then alter it so the order does not matter.

Going back to our pool ball example, let's say we just want to know which 3 pool balls are chosen, not the order.

We already know that 3 out of 16 gave us 3,360 permutations.

But many of those are the same to us now, because we don't care what order!

For example, let us say balls 1, 2 and 3 are chosen. These are the possibilites:

So, the permutations have 6 times as many possibilites.

In fact there is an easy way to work out how many ways "1 2 3" could be placed in order, and we have already talked about it. The answer is:

(Another example: 4 things can be placed in 4! = 4 × 3 × 2 × 1 = 24 different ways, try it for yourself!)

So we adjust our permutations formula to reduce it by how many ways the objects could be in order (because we aren't interested in their order any more):

That formula is so important it is often just written in big parentheses like this:

It is often called "n choose r" (such as "16 choose 3")

And is also known as the Binomial Coefficient.

Notation

All these notations mean "n choose r":

Just remember the formula:

Example: Pool Balls (without order)

So, our pool ball example (now without order) is:

= 560

So choosing 3 balls out of 16, or choosing 13 balls out of 16, have the same number of combinations:

In fact the formula is nice and symmetrical:

Also, knowing that 16!/13! reduces to 16×15×14, we can save lots of calculation by doing it this way:

= 560

Pascal's Triangle

We can also use Pascal's Triangle to find the values. Go down to row "n" (the top row is 0), and then along "r" places and the value there is our answer. Here is an extract showing row 16:

1 14 91 364 ...
1 15 105 455 1365 ...
1 16 120 560 1820 4368 ...

1. Combinations with Repetition

OK, now we can tackle this one ...

Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla.

We can have three scoops. How many variations will there be?

Let's use letters for the flavors: {b, c, l, s, v}. Example selections include

{c, c, c} (3 scoops of chocolate)

{b, l, v} (one each of banana, lemon and vanilla)

{b, v, v} (one of banana, two of vanilla)

(And just to be clear: There are n=5 things to choose from, we choose r=3 of them,
order does not matter, and we can repeat!)

Now, I can't describe directly to you how to calculate this, but I can show you a special technique that lets you work it out.

Think about the ice cream being in boxes, we could say "move past the first box, then take 3 scoops, then move along 3 more boxes to the end" and we will have 3 scoops of chocolate!

So it is like we are ordering a robot to get our ice cream, but it doesn't change anything, we still get what we want.

We can write this down as (arrow means move, circle means scoop).

In fact the three examples above can be written like this:

{c, c, c} (3 scoops of chocolate):

{b, l, v} (one each of banana, lemon and vanilla):

{b, v, v} (one of banana, two of vanilla):

So instead of worrying about different flavors, we have a simpler question: "how many different ways can we arrange arrows and circles?"

Notice that there are always 3 circles (3 scoops of ice cream) and 4 arrows (we need to move 4 times to go from the 1st to 5th container).

So (being general here) there are r + (n−1) positions, and we want to choose r of them to have circles.

This is like saying "we have r + (n−1) pool balls and want to choose r of them". In other words it is now like the pool balls question, but with slightly changed numbers. And we can write it like this:

Interestingly, we can look at the arrows instead of the circles, and say "we have r + (n−1) positions and want to choose (n−1) of them to have arrows", and the answer is the same:

So, what about our example, what is the answer?

There are 35 ways of having 3 scoops from five flavors of icecream.

In Conclusion

Phew, that was a lot to absorb, so maybe you could read it again to be sure!

But knowing how these formulas work is only half the battle. Figuring out how to interpret a real world situation can be quite hard.

But at least you now know the 4 variations of "Order does/does not matter" and "Repeats are/are not allowed":

708, 1482, 709, 1483, 747, 1484, 748, 749, 1485, 750

# data[].attributes.attachment_link.description
In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

# data[].attributes.body
This link provides a quick explanation of the Fundamental Counting Principle.

# data[].attributes.attachment_link.readable_text
The fundamental counting principle states that if there are ways to do one thing, and ways to do another thing, then there are ways to do both things.

Example 1:

possible outfits:

Example 2:

Suppose you roll a -sided die and draw a card from a deck of cards. There are possible outcomes on the die, and possible outcomes from the deck of cards. So, there are a total of

possible outcomes of the experiment.

The counting principle can be extended to situations where you have more than choices. For instance, if there are ways to do one thing, ways to a second thing, and ways to do a third thing, then there are ways to do all three things.

# data[].attributes.attachment_link.description
The fundamental counting principle states that if there are p ways to do one thing, and q ways to do another thing, then there are p × q ways to do both things.

# data[].attributes.body
Follow this link for a quick explanation of the term asymptote.

# data[].attributes.attachment_link.description
An asymptote is a line that a curve approaches, as it heads towards infinity