The binomial distribution

Transcription

1 The binomial distribution The coin toss - three coins The coin toss - four coins The binomial probability distribution Rolling dice Using the TI nspire Graph of binomial distribution Mean & standard deviation 1

2 The coin toss - three coins Three coins are tossed. What is the probability distribution of X (the number of heads?) H Heads & tails are equally likely. All outcomes are equally likely. T H T H T H T H T H T H T HHH HHT HTH HTT THH THT TTH TTT Pr(X=0)= 1 8 Pr(X=1)= 3 8 Pr(X=2)= 3 8 Pr(X=3)= 1 8 2

3 The binomial distribution The coin toss is an eample of a Bernoulli sequence. Repeated trials where only two distinct outcomes (success or failure) are possible. Trials are independent; each trial has the same probability of a successful outcome. The binomial distribution describes the chances of ge ing each possible value of X; the number of successful outcomes from the number of trials. The binomial distribution is an eample of a discrete probability distribution. Coin toss, dice roll or yes / no options will follow a binomial distribution. 3

4 The coin toss - four coins What if there were four coins? There are now 16 (2 4 ) possible outcomes. There are now 6 possible outcomes with two heads. HHTT HTHT HTTH THHT THTH Pr(X=2)= 6 16 = 3 8 TTHH There are si different ways that the two heads can be arranged in the four places. 4

5 The binomial probability distribution This can be found using the idea of Combinations. There are si combinations that can be made by choosing two from four objects. 4 C = 4! 2 2!2! = 24 4 = 6 Pr(X =2)= 4 C = 6 16 Number of trials Number of successes (heads) Number of failures (tails) Pr(X = )= n C p ( ) ( 1 p) n Number of possible arrangements Chance of failure (tails) Chance of success (heads) 5

6 Pascal s triangle (0 trials) 1 trial 2 trials 3 trials 4 trials 5 trials X=0 X=1 X=2 X=3 X=4 X=5 6

7 The binomial probability distribution Four coins: The probability distribution for the number of heads has five possible values (0-4). Pr(X = 0)= 4 C Pr(X =1)= 4 C Pr(X =2)= 4 C Pr(X = 3)= 4 C Pr(X = 4)= 4 C = 1 16 = 4 16 = = 6 16 = 3 8 = 4 16 = 1 4 = 1 16 This distribution is symmetrical when the chances of success (head) or failure (tail) are equal. The sum of all probabilities must be equal to one. 7

8 Rolling dice Four dice are rolled. What is the distribution that describes the number of sies rolled? Pr(X = 0)= 4 C Pr(X =1)= 4 C Pr(X =2)= 4 C Pr(X = 3)= 4 C Pr(X = 4)= 4 C = = 48.2% = = 38.6% = =11.6% 5 = =1.5% 1 = = 0.08% 8

9 How many trials? Each roll of the die has 1/6 chance of rolling a 6. How many trials (rolls) are needed to have a 90% probability of rolling a 6 at least once? At least one 6 rolled in trials includes all possible options epect for rolling none. Pr(X = 0)= Pr(X > 0)= = = = 5 6 (Or solve on a CAS calculator.) log(0.1)= log 5 6 = log(0.1) log(5 / 6) 13 (Always round up) 9

10 How many trials? Each roll of the die has 1/6 chance of rolling a 6. How many rolls are needed to have a 90% probability of rolling a 6 at least twice? Pr(X 2)= 90% Pr(X <2)=10% = Pr(X = 0)+Pr(X =1) 10% = n C % = = 5 6 n 0 5 n 6 +n 1 6 n +n n C n n n 1 6 (Solve on a CAS calculator.) n 22 (Always round up) 10

11 Using the TI-nSpire CAS calculators and spreadsheets have a built in function for binomial distributions. For eample Pr (X=0) from four dice: Menu > 5: Probability > 5: Distributions > D: Binomial Pdf The cumulative distribution (Cdf) is used to find the sum of all probabilities above or below a value of X eg Pr (X 3). Pr(X = 0)= 4 C = 48.2% 11

12 Mean & standard deviation A probability distribution has a mean (epected value) and standard deviation (average variation from mean). For the binomial distribution: E(X )= np Var(X )= np(1 p ) Four coin toss: E(X )= 4 0.5=2 Var(X )= =1 SD(X )= 1=1 Rolling four dice: E(X )= = 2 3 = 0.67 The highest variance is when there is 50% chance of success. Var(X )= = 20 SD(X )= 0.56 = =

13 Graphs of probability distributions This distribution is for ten trials. If the chance of success is 50%, the distribution is symmetrical. 40% Pr (X=) Pr = % 20% 10% 0%

14 Graphs of probability distributions If the chance of success is less than 50%, the distribution is skewed towards the right. The epected value of X is less than the middle value. 40% Pr (X=) Pr = % 20% 10% Values too small to see. 0%

15 Graphs of probability distributions If the chance of success is more than 50%, the distribution is skewed towards the left. The epected value of X is greater than the middle value. 40% Pr (X=) Pr = % 20% 10% 0%