MATH 1680 - Section 6.2 - The Binomial Probability Distribution
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Formula for computing probabilities of binomial experiments
P(x) = (ₙCₓ)(p^x)(1-p)^(n-x) where p is the prob of success
HOW TO SOLVEl According to a report, 52.3% of murders are committed with a firearm. (a) If 400 murders are randomly selected, how many would we expect to be committed with a firearm? (b) Would it be unusual to observe 238 murders by firearm in a random sample of 400 murders? Why? (a) We would expect 209209 to be committed with a firearm. (b) Choose the correct answer below.
a) find the mean by multiply np b) find the two standard deviations. if 238 is less than two dstandard devations plus the mean,then its usual
11) As n increases, describe what happens to the shape of a binomial probability distribution.
as n, increases, the sahpe of a binoamil probablity graph morphs into a symmetrical shape/bell shape
10) What is the shape of the binomial probability distribution if p< 0.5, if 0.5 = p , and if p > 0.5 ?
p<0.5, skewed right p=0.5, symmetrical or bell shaped p>0.5, skewed left
Is the following a binomial experiment? If so, state the number of trials, prob of success, prob of failure, and the possible values of the random variable X B) According to a recent Harris Poll, 28% of Americans state that chocolate is their favorite flavor of ice cream. Suppose a simple random sample of size 10 is obtained and the number of Americans who choose chocolate as their favorite ice cream flavor is recorded.
yes, 10 trials, p of success is .28, 1-p is .71, X=0,1,2,3,..,10
Formulas for the mean or expected value
μₓ = np
formula for the standadrd deviation of a binomial random varaible
σₓ=√np(1-p)
how to solve: Use n=6 and p=0.2 to complete parts (a) through (d) below. (a) Construct a binomial probability distribution with the given parameters. x P(x) 0 .2621 1 .3932 2 .2458 3 .0819 4 .0154 5 .0015 6 .0001
P(x)=nCxpx(1−p)n−x, use that formula to compute p(x)
What do n, p, and 1 - p represent when working with a binomial probability distribution?
-there are n independent trials of the experiment -let p denote the probabilty of success so that 1-p is the probability of failure -let x be a binomial random variable that denotes the number of successes in n independent trails of the experiment. so 0≤x≤1
In the formula, what would 0.07 and 1 represent? eg (ₙCₓ)(.07^1)(1-p)^(n-x)
.07 is the p(sucess) and one is the number of successes
6) In the formula, what do 0.93 and 3 represent? P(x) = (ₙCₓ)(p^x)(1-.93)³
.93 is the p(failure). the exponent three is the number of failures
What is the required criteria to label an experiment as a binomial experiment
1) the experiment is performed a fixed number of times. each repetition of the experiment is called a trial. 2) the trials are independent. this means the outcome of one trail will not affect the outcome of the other trials. 3) for each trial, there are two mutually exclusive (or disjoint) otucomes, sucess or failure 4) the probabily of cusses is fixed for each trial of the experiment
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A binomial experiment is performed a fixed number of times. What is each repetition of the experiment called? A. Each repetition of the experiment is called a mean. B. Each repetition of the experiment is called a success. C. Each repetition of the experiment is called a trial. D. Each repetition of the experiment is called a binomial random variable.
Each repetition of the experiment is called a trial.
In the formula, what would ₄C₁ represent?
FOUR IS THE NUMBER OF WAYS TO OBTAIN ONE success in four trials of an experiment
HOW TO sOLVE: According to an airline, flights on a certain route are on time 80% of the time. Suppose 13 flights are randomly selected and the number of on-time flights is recorded. (a) Explain why this is a binomial experiment. (b) Determine the values of n and p. (c) Find and interpret the probability that exactly 8 flights are on time. (d) Find and interpret the probability that fewer than 8 flights are on time. (e) Find and interpret the probability that at least 8 flights are on time. (f) Find and interpret the probability that between 6 and 8 flights, inclusive, are on time.
GO TO STATCRUNCH BINOMIAL CALCULATOR
According to an almanac, 70% of adult smokers started smoking before turning 18 years old. (a) Compute the mean and standard deviation of the random variable X, the number of smokers who started smoking before 18 based on a random sample of 200 adults. (a) μx=140 σx=6.5 (b) Interpret the mean.
It is expected that in a random sample of 200 adult smokers, 140 will have started smoking before turning 18.
If X is a binomial random variable that denotes the number of successes in n independent trials of an experiment, what are the possible values of X?
X = 1,2,3,4,...,10
binomial probability distribution
a discrete probability distribution that describes probabilities for experiments in which there are two mutually exclusive (disjoint) outcomes: success and failure
How to solve: A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n=10, p=0.65, x= 7 compute p(7) =
go to binomial calculator on statcrunch
In a certain state, 43% of adults indicated that sausage is their favorite pizza. Suppose a simple random sample of adults in the state of size 25 is obtained and the number of adults who indicated that sausage is their favorite pizza was 12. What are values of the parameters n, p, and x in the binomial probability experiment?
n = 25 p = .43 x= 12
Is the following a binomial experiment? If so, state the number of trials, prob of success, prob of failure, and the possible values of the random variable X C) A probability experiment in which three cards are drawn from a deck without replacement and the number of aces is recorded
no, because the trials are dependent
13) Explain how to determine if an observation in a binomial experiment is unusual.
recall that the emperical rule states that in a bell-shaped distriubtion, about 95% of all observations lie within two standard devations of the mean. that is, about 95% of the boseravtions lie between μ-2σ and μ+2σ. any observation that lies outside this interval may be considered unusual because the observation occurs less than 5% of the time
Statcrunch to graph binomial prob distribution
stat>calculator>binomial>insert n: ?>insert p: ?> computer when p(X) is too small statcrunch wont graph it but you can see its value on (x = ?)
statcrunch steps for binomial probability distribution
stat>calculators>binomials> n: (input n)> p(x: (input x) > compute *choose between = > < ≤≥ you can also choose p(?<x<?) by clicking between
State the binomial probability distribution function
the probability of obtaining x successes in n independent trials of a binomial experiment is given by: P(x) = (ₙCₓ)(p^x)(1-p)^(n-x) x= 0,1,2...,n where p is the probability of success
12) Under what conditions will a binomial probability distribution be approximately bell-shaped?
the probablity distribution is bell shaped if np(1-p)≥10
HOW TO SOLVE: According to CTIA, 55% of all U.S. households are wireless-only households. In a simple random sample of 500 households, determine the mean and standard deviation number of wireless-only households
to fidn the mean, multiply n by p so 500 by .55 to find the std deviation, find the square root of np times one minus p
Is the following a binomial experiment? If so, state the number of trials, prob of success, prob of failure, and the possible values of the random variable X A) An experiment in which a basketball player who historically makes 80% of his free throws is asked to shoot three free throws and the number of free throws made is recorded
yes it is a binomial experiment: 3 trials, P is .8, 1-P is .2, and X = 0,1,2,3
Determine if the following probability experiment represents a binomial experiment. If not, explain why. If the probability experiment is a binomial experiment, state the number of trials, n. Seven cards are selected from a standard 52-card deck without replacement. The number of fives selected is recorded.
No, because the trials of the experiment are not independent since the probability of success differs from trial to trial.
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