Binomial Distributions (aka Bernouli’s Trials)

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Binomial Distributions Chapter 8 (aka Bernouli’s Trials)

Transcript of Binomial Distributions (aka Bernouli’s Trials)

Page 1: Binomial Distributions (aka Bernouli’s Trials)

Binomial Distributions

Chapter 8

(aka Bernouli’s Trials)

Page 2: Binomial Distributions (aka Bernouli’s Trials)

Binomial Distribution

•an important class of ___________ probability distributions, which occur under the following __________ _____________

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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 ____________

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• 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.

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• 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 ____________ ______________ _______________.

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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.

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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

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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

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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?

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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?

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On the Calculator

•use the binompdf function under the DISTR menu:

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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

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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

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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.

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We can also find the population parameters for Binomial Distribution using the following:•Population Parameters of a Binomial Distribution

•Mean:

•Standard deviation:

•Variance:

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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

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•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?

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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 = ___________

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(the hardest to remember)

•For at least probability:

• use

• L =

•Enter in calculator:

•This gives the probability at the at least number.

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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?

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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?