Binomial Distributions

Transcription

1 Binomial Distributions (aka Bernouli s Trials) Chapter 8

2 Binomial Distribution an important class of probability distributions, which occur under the following

3 Binomial Setting (1) There is a number n of observations. (2) There are possible outcomes success or failure (3) The probability of, called p, is the for each observation. (4) The n observations are : knowing the result of one observation tells you nothing about the others. And the variables are, or

4 Binomial distribution probability model describes the of success in a number of trials. For Binomial distribution we will look at the probability of getting an event with: n = k = p = (1 p)=

5 If X is a binomial random variable, it is said to have a distribution, and is denoted as. If data are produced in a binomial setting then the random variable X = number of successes is called a.

6 Are the following in the binomial setting? If so, what does n, k, p and 1-p equal? Blood type inherited. If both parents carry genes for both O and A blood types each child has a probability of 0.25 of getting 2 O genes and so having blood type O. Different children inherit independently of each other. The number of O blood types among 5 children is the count x in 5 observations. Deal 10 cards from a shuffled deck and count the numbers x or red cards. There are 10 observations and red is a success.

7 Binomial Coefficient also called a, is the number of ways to arrange k successes in n observations. It is written and is read as n choose k. The value is given by the formula

8 Probability Formula: If X is a binomial random variable with parameters n and p, then for any k in n the binomial probability of k is

9 Example Suppose each child born to Jay and Kay has probability 0.25 of having blood type O. If Jay and Kay have 5 children, what is the probability that exactly 2 of them have type O blood?

10 Example If the probability that the Panthers will win a game is 0.2, what is the probability that they a) win exactly 2 out of their next 3 games? b) win at most 1 out of their next 5 games? c) win a least four of their next 5 games?

11 On the Calculator use the binompdf function under the DISTR menu:

12 Probability Distribution Function The (pdf) assigns a probability to each value of X Example: X P(X)

13 Cumulative Distribution Function The ( ) calculates the sum of the probabilities up to X. X P(X) F(X) P(X 0) P(X 1) P(X 2) P(X 3) P(X 4) P(X 5) 1.0

14 Example If the probability that the panthers will win is 0.05 (they may need a new coach), create a probability distribution table to the next 4 games that they will play.

15 We can also find the population parameters for Binomial Distribution using the following: Population Parameters of a Binomial Distribution Mean: Standard deviation: Variance:

16 Rule of Thumb When n is large the distribution of X is approximately normal so we can use to estimate probabilities. As a rule of thumb we use normal approximation when and

17 Find the mean, variance and standard deviation of the following: 1) A child born has probability of 0.25 of having blood type O. If five children are born, what is the probability that exactly two of them will have type O blood. 2) If the probability that the Panthers will win a game is 0.2, what is the probability that they will win exactly 2 out of their next 5 games?

18 We can do the binomial calculation in the calculator by using the binomial cdf or pdf commands. For exact probability: Use It gives an number (the answer) For at most probability: use It gives p =

19 (the hardest to remember) For at least probability: use L = Enter in calculator: This gives the probability at the at least number.

20 Roll a die 5 times. What is the probability of getting a 4 Exactly once? Exactly three times? At most 3 times? At least 3 times?

21 A certain tennis player makes a successful serve 70% of the time. Assume that each serve is independent of the others, If she serves 6 times, what is the probability that she gets Exactly 4 serves in? All 6 serves in? At least 4 serves in? No more than 4 serves in?