Binomial distribution

When events are independant:

The Binomial Distribution helps us determine the probability of a string of independent 'coin flip like events'.

The probability mass function associated with the binomial distribution is of the following form:

where n is the number of events, x is the number of "successes", and p is the probability of "success".

We can now use this distribution to determine the probability of things like:

• The probability of 3 heads occurring in 10 flips.

• The probability of observing 8 or more heads occurring in 10 flips.

• The probability of not observing any heads in 20 flips.

Quizz:

With 10 coin flips, how many combinations of way can I have exaclty 4 heads as a result.

Answer = the fraction above.

The formula is:

This keeps track of the total number of ways we can get k heads for n coin flips.

Exercise:

With a coin that isn't fair:

We can multiply the number of ways 4 heads will show up (which is 5 - see truth table), with the prob of heads * 4 (we want to know P(#head =4) so that's (0.8)^4, * same thing with tails, thats (0,2)^1 = 0.4096.

So the formula for the probability is:

p^k (first part of the formula) = probability associated with the number of heads

(1-p)^(n-k) (2nd part) = probability associated with the number of tails.

You can apply this formula for anything that has two outcomes.