Binomial Distributions
Chapter 8
(aka Bernouli’s Trials)
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 ____________
• For Binomial distribution we will look at the probability of getting an event with:
• n =
• k =
• p =
• (1 – p)=
Binomial distribution probability model describes the
__________ of success in a _________ number of trials.
• 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 nobservations. 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?
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