Heads or Tails?

start with the binomial probability distribution.

also, Pascal's triangle has the answers for the number of ways it can happen.

example: Probability of getting 5 heads in 10 flips? Using a fair coin.

Row 10 is this: 1,10,45,120,210,252,210,120,45,10,1 (row highlighted in below photo)

it means this is how many Heads there are. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10 (heads or successes)

the 6th element in the row = 252 ways.

the probability is then:

252 / 2^10 or

252 / 1024 or about 24.6%

1r. using R for the number of ways (where order does not matter) from N to choose X.

choose(10,5) means we have 10 and want to choose 5

in other words 10 choose 5

2r. R code for the binomial distribution.

for the probability mass function (pmf). exactly X Heads (successes) in N flips.

call function with example: data(10, 1/2, 0, 10)

data(number of trials, probability of success at each trial, min success, max success)

example: data(10, 1/2, 5, 5) will return the probability for just 5 successes

cumulative probability has been added for a range

3r. R code for the cumulative distribution function and inverse (at least X) .

exactly X Heads or less (successes) in N flips. (pmf)

call function with example: data(10, 1/2, 0, 10)

data(number of trials, probability of success at each trial, min success, max success)

example: data(10, 1/2, 5, 5) will return the probability for just 5 successes or less

4r. R code to produce up to row x for Pascal's triangle rows.

Index starts at 1 which is row 0

5r. R code to produce a specific row for Pascal's triangle.

Returns a matrix showing the number of ways for 'choose k' from n