Binomial Distributions (aka Bernouli s Trials) Chapter 8
Binomial Distribution an important class of probability distributions, which occur under the following
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
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)=
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.
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.
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
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
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?
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?
On the Calculator use the binompdf function under the DISTR menu:
Probability Distribution Function The (pdf) assigns a probability to each value of X Example: X 0 1 2 3 4 5 P(X) 0.237 0.396 0.264 0.088 0.015 0.001
Cumulative Distribution Function The ( ) calculates the sum of the probabilities up to X. X 0 1 2 3 4 5 P(X) 0.237 0.396 0.264 0.088 0.015 0.001 F(X) P(X 0) 0.237 P(X 1) 0.633 P(X 2) 0.897 P(X 3) 0.984 P(X 4) 0.999 P(X 5) 1.0
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.
We can also find the population parameters for Binomial Distribution using the following: Population Parameters of a Binomial Distribution Mean: Standard deviation: Variance:
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
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?
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 =
(the hardest to remember) For at least probability: use L = Enter in calculator: This gives the probability at the at least number.
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?
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?