To find a missing number in a Sequence, first we must have a Rule

  1. 6x + 3(0) – 1 = 6x – 1. Simplified form: 6x + 3h – 1. Value at h = 0: 6x – 1. That's pretty much all there is to 'operations on functions' until you get to function composition. Don't let the notation for this topic worry you; it means nothing more than exactly what it.
  2. Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.

Sequence

The value x = 0.5 is an atom of the distribution of X, thus, the corresponding conditional distribution is well-defined and may be calculated by elementary means (the denominator does not vanish); the conditional distribution of Y given X = 0.5 is uniform on (2/3, 1). Measure theory leads to the same result.
A Sequence is a set of things (usually numbers) that are in order. Easy unzip 1 5.
Each number in the sequence is called a term (or sometimes 'element' or 'member'), read Sequences and Series for a more in-depth discussion.

Finding Missing Numbers

To find a missing number, first find a Rule behind the Sequence.
Sometimes we can just look at the numbers and see a pattern:

Example: 1, 4, 9, 16, ?

Answer: they are Squares (12=1, 22=4, 32=9, 42=16, ..)

Rule: xn = n2

Sequence: 1, 4, 9, 16, 25, 36, 49, .. Barcode maker 1 0.
Did you see how we wrote that rule using 'x' and 'n' ?

xn means 'term number n', so term 3 is written x3

And we can calculate term 3 using:

x3 = 32 = 9

We can use a Rule to find any term. For example, the 25th term can be found by 'plugging in' 25 wherever n is.

x25 = 252 = 625

How about another example:

Example: 3, 5, 8, 13, 21, ?

After 3 and 5 all the rest are the sum of the two numbers before,
That is 3 + 5 = 8, 5 + 8 = 13 etc, which is part of the Fibonacci Sequence:
3, 5, 8, 13, 21, 34, 55, 89, ..
Which has this Rule:
Allmymusic 3 0 1 5 X 4

Rule: xn = xn-1 + xn-2

Now what does xn-1 mean? It means 'the previous term' as term number n-1 is 1 less than term number n.
And xn-2 means the term before that one.
Let's try that Rule for the 6th term:

x6 = x6-1 + x6-2

x6 = x5 + x4

So term 6 equals term 5 plus term 4. We already know term 5 is 21 and term 4 is 13, so:

x6 = 21 + 13 = 34

Many Rules

One of the troubles with finding 'the next number' in a sequence is that mathematics is so powerful we can find more than one Rule that works.

What is the next number in the sequence 1, 2, 4, 7, ?

Here are three solutions (there can be more!):

Solution 1: Add 1, then add 2, 3, 4, ..
So, 1+1=2, 2+2=4, 4+3=7, 7+4=11, etc..
Rule: xn = n(n-1)/2 + 1
Sequence: 1, 2, 4, 7, 11, 16, 22, ..
(That rule looks a bit complicated, but it works)
Solution 2: After 1 and 2, add the two previous numbers, plus 1:
Rule: xn = xn-1 + xn-2 + 1
Sequence: 1, 2, 4, 7, 12, 20, 33, ..
Solution 3: After 1, 2 and 4, add the three previous numbers
Rule: xn = xn-1 + xn-2 + xn-3
Sequence: 1, 2, 4, 7, 13, 24, 44, ..
So, we have three perfectly reasonable solutions, and they create totally different sequences.

Which is right? They are all right.

And there are other solutions ..
.. it may be a list of the winners' numbers .. so the next number could be .. anything!

Simplest Rule

When in doubt choose the simplest rule that makes sense, but also mention that there are other solutions.

Finding Differences

Sometimes it helps to find the differences between each pair of numbers .. this can often reveal an underlying pattern.
Here is a simple case:
The differences are always 2, so we can guess that '2n' is part of the answer.
Let us try 2n:
The last row shows that we are always wrong by 5, so just add 5 and we are done:

Rule: xn = 2n + 5

OK, we could have worked out '2n+5' by just playing around with the numbers a bit, but we want a systematic way to do it, for when the sequences get more complicated.

Second Differences

In the sequence {1, 2, 4, 7, 11, 16, 22, ..} we need to find the differences ..
.. and then find the differences of those (called second differences), like this:
The second differences in this case are 1.

With second differences we multiply by n22

In our case the difference is 1, so let us try just n22:
n:12345
Terms (xn):124711
n22:0.524.5812.5
Wrong by:0.50-0.5-1-1.5
We are close, but seem to be drifting by 0.5, so let us try: n22 − n2
Wrong by 1 now, so let us add 1:
n22 − n2 + 1124711
Wrong by:00000
We did it!
The formula n22 − n2 + 1 can be simplified to n(n-1)/2 + 1
Activedock. So by 'trial-and-error' we discovered a rule that works:

Rule: xn = n(n-1)/2 + 1

Sequence: 1, 2, 4, 7, 11, 16, 22, 29, 37, ..

Other Types of Sequences

Read Sequences and Series to learn about:
And there are also:
And many more!
In truth there are too many types of sequences to mention here, but if there is a special one you would like me to add just let me know.
'Bi' means 'two' (like a bicycle has two wheels) ..
.. so this is about things with two results.
Tossing a Coin:
We say the probability of the coin landing H is ½
And the probability of the coin landing T is ½
Throwing a Die:
We say the probability of a four is 1/6 (one of the six faces is a four)
And the probability of not four is 5/6 (five of the six faces are not a four)
Note that a die has 6 sides but here we look at only two cases: 'four: yes' or 'four: no'

Let's Toss a Coin!

Toss a fair coin three times .. what is the chance of getting two Heads?
Tossing a coin three times (H is for heads, T for Tails) can get any of these 8 outcomes:
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT

Which outcomes do we want?

'Two Heads' could be in any order: 'HHT', 'THH' and 'HTH' all have two Heads (and one Tail).
So 3 of the outcomes produce 'Two Heads'.

What is the probability of each outcome?

Each outcome is equally likely, and there are 8 of them, so each outcome has a probability of 1/8
So the probability of event 'Two Heads' is:
Number of
outcomes we want		Probability of
each outcome
3 × 1/8 = 3/8
So the chance of getting Two Heads is 3/8
We used special words:

3 Heads, 2 Heads, 1 Head, None

The calculations are (P means 'Probability of'):
We can write this in terms of a Random Variable, X, = 'The number of Heads from 3 tosses of a coin':
And this is what it looks like as a graph:

It is symmetrical!

Making a Formula

Now imagine we want the chances of 5 heads in 9 tosses: to list all 512 outcomes will take a long time!
So let's make a formula.
In our previous example, how can we get the values 1, 3, 3 and 1 ?
Well, they are actually in Pascal’s Triangle !
Can we make them using a formula?
Sure we can, and here it is:

It is often called 'n choose k'
You can read more about it at Combinations and Permutations.
Let's try it:

Example: with 3 tosses, what are the chances of 2 Heads?

Allmymusic 3 0 1 5 X 4 5 Bolt Pattern

We have n=3 and k=2:
=3×2×12×1 × 1
So there are 3 outcomes that have '2 Heads'
(We knew that already, but now we have a formula for it.)
Let's use it for a harder question:

Example: with 9 tosses, what are the chances of 5 Heads?

We have n=9 and k=5:
=9×8×7×6×5×4×3×2×15×4×3×2×1 × 4×3×2×1
So 126 of the outcomes will have 5 heads
And for 9 tosses there are a total of 29 = 512 outcomes, so we get the probability:
Number of
outcomes we want		Probability of
each outcome
126 × 1512 = 126512
So:
P(X=5) = 126512 = 0.24609375
About a 25% chance.
(Easier than listing them all.)

Bias!

So far the chances of success or failure have been equally likely.
But what if the coins are biased (land more on one side than another) or choices are not 50/50.

Example: You sell sandwiches. 70% of people choose chicken, the rest choose something else.

What is the probability of selling 2 chicken sandwiches to the next 3 customers?

This is just like the heads and tails example, but with 70/30 instead of 50/50.
Let's draw a tree diagram:
The 'Two Chicken' cases are highlighted.
The probabilities for 'two chickens' all work out to be 0.147, because we are multiplying two 0.7s and one 0.3 in each case. In other words
0.147 = 0.7 × 0.7 × 0.3
Or, using exponents:
= 0.72 × 0.31
The 0.7 is the probability of each choice we want, call it p
The 2 is the number of choices we want, call it k
And we have (so far):
= pk × 0.31
The 0.3 is the probability of the opposite choice, so it is: 1−p
The 1 is the number of opposite choices, so it is: n−k
Which gives us:
= pk(1-p)(n-k)
Where

Example: (continued)

So we get:
=0.72(0.3)(1)
=0.147
which is what we got before, but now using a formula
Now we know the probability of each outcome is 0.147
But we need to include that there are three such ways it can happen: (chicken, chicken, other) or (chicken, other, chicken) or (other, chicken, chicken)

Example: (continued)

The total number of 'two chicken' outcomes is:
=3×2×12×1 × 1
And we get:
Number of
outcomes we want		Probability of
each outcome
3 × 0.147 = 0.441
So the probability of event '2 people out of 3 choose chicken' = 0.441
OK. That was a lot of work for something we knew already, but now we have a formula we can use for harder questions.

Example: Sam says '70% choose chicken, so 7 of the next 10 customers should choose chicken' .. what are the chances Sam is right?

So we have:
And we get:
=0.77(0.3)(3)
That is the probability of each outcome.
And the total number of those outcomes is:
=10×9×8×7×6×5×4×3×2×17×6×5×4×3×2×1 × 3×2×1
=120

0 1 Factorial

And we get:
Number of
outcomes we want		Probability of
each outcome
120 × 0.0022235661 = 0.266827932
So the probability of 7 out of 10 choosing chicken is only about 27%
Moral of the story: even though the long-run average is 70%, don't expect 7 out of the next 10.

Putting it Together

Now we know how to calculate how many:
n!k!(n-k)!
And the probability of each:
pk(1-p)(n-k)
When multiplied together we get:
Probability of k out of n ways:
P(k out of n) = n!k!(n-k)!pk(1-p)(n-k)
The General Binomial Probability Formula
Important Notes:

Quincunx

Have a play with the Quincunx (then read Quincunx Explained) to see the Binomial Distribution in action.

Throw the Die

A fair die is thrown four times. Calculate the probabilities of getting:
In this case n=4, p = P(Two) = 1/6
X is the Random Variable ‘Number of Twos from four throws’.

Allmymusic 3 0 1 5 X 4 5 Solar Post Caps

Substitute x = 0 to 4 into the formula:
P(k out of n) = n!k!(n-k)! pk(1-p)(n-k)
Like this (to 4 decimal places):
Summary: 'for the 4 throws, there is a 48% chance of no twos, 39% chance of 1 two, 12% chance of 2 twos, 1.5% chance of 3 twos, and a tiny 0.08% chance of all throws being a two (but it still could happen!)'
This time the graph is not symmetrical:

It is not symmetrical!
It is skewed because p is not 0.5

Sports Bikes

Your company makes sports bikes. 90% pass final inspection (and 10% fail and need to be fixed).
What is the expected Mean and Variance of the 4 next inspections?
First, let's calculate all probabilities.
X is the Random Variable 'Number of passes from four inspections'.
Substitute x = 0 to 4 into the formula:
P(k out of n) = n!k!(n-k)! pk(1-p)(n-k)
Like this:
Summary: 'for the 4 next bikes, there is a tiny 0.01% chance of no passes, 0.36% chance of 1 pass, 5% chance of 2 passes, 29% chance of 3 passes, and a whopping 66% chance they all pass the inspection.'

Mean, Variance and Standard Deviation

Let's calculate the Mean, Variance and Standard Deviation for the Sports Bike inspections.
There are (relatively) simple formulas for them. They are a little hard to prove, but they do work!
The mean, or 'expected value', is:
μ = np
For the sports bikes:
μ = 4 × 0.9 = 3.6
So we can expect 3.6 bikes (out of 4) to pass the inspection.
Makes sense really .. 0.9 chance for each bike times 4 bikes equals 3.6
The formula for Variance is:
Variance: σ2 = np(1-p)
And Standard Deviation is the square root of variance:
σ = √(np(1-p))

Allmymusic 3 0 1 5 X 4 5 Picture Frame

For the sports bikes:
Variance: σ2 = 4 × 0.9 × 0.1 = 0.36
Standard Deviation is:
σ = √(0.36) = 0.6
Note: we could also calculate them manually, by making a table like this:
X P(X) X × P(X) X2 × P(X)
00.0001 0 0
10.0036 0.00360.0036
20.0486 0.0972 0.1944
30.2916 0.8748 2.6244
40.6561 2.6244 10.4976
SUM: 3.613.32
The mean is the Sum of (X × P(X)):
μ = 3.6
The variance is the Sum of (X2 × P(X)) minus Mean2:
Variance: σ2 = 13.32 − 3.62 = 0.36
Standard Deviation is:

0.1 As A Fraction

σ = √(0.36) = 0.6
And we got the same results as before (yay!)

Summary