Stone Studio

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1 When can you use the binomial formula?

We want to know how many times something does or doesn’t happen e.g. rolling a
die multiple times:

• can use binomial if we want to know “number of sixes”
• can’t use binomial if we want to know “sum of rolls”

2 Where did that binomial coefficient come from?

Say you have 8 coins, 4 different heads: H1 H2 H3 H4; and 4 different tails: T1 T2 T3
T4. How many ways are there of arranging these 8 coins?

Answer: 8 ways of picking the first coin, then 7 of picking the second given the
first, then 6 of the third…

If there are 8 different objects, then there are 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 ways of
arranging them

We write this as 8! (pronounced “eight factorial”)

Now, if we don’t have 8 different objects: the four head are the same. How many
times are we counting each set of four heads?

The four heads (H1, H2, H3, H4) could be in any of 4! different orders. But
the four tails could be in any of 4! different orders. Then number of ways
is:

3 The binomial formula

The is often expressed in a shorthand notation:

or in calculators

4 Examples

4.1 Counting sixes

I roll a die ten times. What’s the probability I get exactly one six?

Use the formula:

4.2 Using the addition rule

I roll a die ten times. What’s the probability I get exactly two sixes or fewer?

Can add up P + P + P, find each of these
probabilities using the binomial formula