The binomial distribution

Throwing five dice Number of sixes 1 2 3 4 5 Tally: Relative frequency

Throwing five dice What is the probability distribution? Number of sixes 1 2 3 4 5 Probability What is the probability distribution?

Tree diagram, two dice

Tree diagram, three dice The probability of all the equivalent paths is the same. We multiply by the number of possible paths.

Two dice: Three dice: “Pascal’s triangle” Number of sixes 1 2 1 2 Probability Three dice: Number of sixes 1 2 3 Probability “Pascal’s triangle”

Five dice: Four dice: Number of sixes 1 2 3 4 Probability 1 2 3 4 Probability Five dice: Number of sixes 1 2 3 4 5 Probability

Probability distribution for the number of sixes in 30 throws.

P(no sixes, five of anything else) Why “binomial”? P(no sixes, five of anything else) P(five sixes)

X = the result of the experiment (number of successes) r = a number in the range 0 to n n = number of trials p = probability of each “success” q = 1-p = probability of each “failure”

Success or failure? With 5 dice if just two are a “six”, the other 3 are “not six”:

A range of outcomes With 20 coins, if we get 1 heads, there must be 19 tails: A cumulative probability is for a range of outcomes: