Business Statistics Chapter 6 - Discrete Probability Distributions
Binomial Probability Formula
P(x)= nCx π^x(1−π)^n−x where pi is the probability or P(success) n is the number of trials x is the random variable C is a combination
A binomial experiment has n=6 trials and a probability of success of pi=0.4 Match the given probabilities with their solutions.
P(x>=4) -> P(4) + P(5) + P(6) P(x<=2) -> P(0) + P(1) + P(2) P(x!=3) -> 1 - P(3) P(x<4) -> P(0) + P(1) + P(2) + P(3)
A binomial experiment consists of tossing 10 coins and observing the number of "heads" that land face-up. Identify any questions from this list that would usually be answered by using a cumulative probability distribution. Select all that apply.
- What is the probability of six or more "heads"? - P(H=6)P(H=7)P(H=8)P(H=9)P(H=10) -What is the chance of four or fewer "heads"? - P(H<=4) = 1 - P(H>4) both of these have more than a single probability unlike the other options
MEAN OF A PROBABILITY DISTRIBUTION
μ=Σ[xP(x)]
variance of binomial distribution
σ^2=nπ(1−π)
VARIANCE OF A PROBABILITY DISTRIBUTION
σ^2=Σ[(x−μ)^2 P(x)]
calculate the mean for the probability distribution shown here. X P(X) -------------- 0 0.1 2 0.3 4 0.4 6 0.2
3.4 μ=Σ(xP(X))= (0*.1)+(2*.3)+(4*.4)+(6*.2)
A binomial distribution has n=12 trials with a probability of success of pi=0.3. Calculate the mean of this binomial distribution.
3.60 u = n*pi u = 12 * .3
PROBABILITY DISTRIBUTION
A listing of all the outcomes of an experiment and the probability associated with each outcome.
DISCRETE RANDOM VARIABLE
A random variable that can assume only certain clearly separated values.
RANDOM VARIABLE
A variable measured or observed as the result of an experiment. By chance, the variable can have different values.
Calculate the standard deviation (rounded to one decimal place) of the probability distribution shown here. X P(X) ----------- 2 0.5 4 0.2 6 0.2 8 0.1
2.1
A bowl contains two red and three black marbles. A marble is randomly selected, its color noted, and returned to the bowl. Let "success" be selecting a red marble. Use the binomial probability formula to find the chance of selecting three red marbles in eight draws.
0.279 x=3, n=8, pi=.4 P(3)=8C3 (0.1)^3 (0.9)^5
Suppose that 40% of households use their cell phones for their home phone. If 12 households are randomly selected, what is the probability that at least three use their cell phones? Assume the binomial distribution applies.
0.917
choose the random variables from this set that are discrete. select all that apply 1. number of drive-thru customers to the bank on a given day 2. the weight of a bag of a dozen apples 3. the number of parking tickets given on campus today 4. the travel time of an airline flight
1, 3
From the following list, choose the random variables that are continuous. Select all that apply
1. The amount of times a light bulb lasts 2. The tire pressure of a tire on a randomly selected automobile
if the standard deviation of probability distribution is 4, what is the variance?
16
expected value
The mean of a probability distribution. a weighted average where the possible values of a random variable are weighted by their corresponding probabilities of occurrence.
Jerry rolls a pair of dice and counts a total of seven dots on the uppermost faces of the dice. Match the feature of this process to the correct term.
experiment - rolling the dice and counting the dots; the experiment is the activity undertaken outcome - getting a 4 on one die and a 3 on the other; a specific result of the experiment event - getting dice that total 7; an event is combination of one or more outcomes random variable - the number of dots showing; a numerical representation of an outcome
the table shown provides binomial probabilities for n=6. If 80% of drivers admit to talking on their cell phone while they drive, what is the probability that half of 6 randomly sampled adults would indicate they talk on their cell phones while driving?
0.082 because when x=3 and 0.80 then the result is 0.082
A software salesman knows that on average he will make one sale for every 10 companies he calls. Let "success" be making a sale. (P(success)=0.10) Use binomial formula to find the chance that he will make two sales if he calls 6 companies.
.098 x=2, n=6, pi=0.1 P(2)= 6C2(0.1)^2 (0.9)^4
It is easier to use a binomial probability table instead of calculating the probability under what circumstances?
When n is large
mean of a binomial distribution
μ = n*pi
BINOMIAL PROBABILITY EXPERIMENT
1. An outcome on each trial of an experiment is classified into one of two mutually exclusive categories—a success or a failure. 2. The random variable is the number of successes in a fixed number of trials. 3. The probability of success is the same for each trial. 4. The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial.
Which of the following is NOT true about binomial probability calculations?
When the x is greater than n, the probability is less than 0. Because x cannot be greater than n and probability cannot be less than 0.
Place the following computational steps in the order in which you would do them to find the standard deviation for a probability distribution.
1. Calculate the mean of the distribution 2. subtract the mean from each value and square the difference 3. multiply each squared difference by its probability 4. sum the products to find the variance 5. take the square root of the variance
Choose the random variables from this set that are discrete. Select all that apply
1. The number of concert tickets sold 2. The number of people riding a bus
CHARACTERISTICS OF A PROBABILITY DISTRIBUTION
1. The probability of a particular outcome is between 0 and 1 inclusive. 2. The outcomes are mutually exclusive. 3. The list of outcomes is exhaustive. So the sum of the probabilities of the outcomes is equal to 1.
Which two of the following statements describe the features of a binomial experiment?
1. The random variable counts the number of successes in a fixed number of trials 2. the probability of success stays the same for each trial
A binomial distribution has 8 trials and a probability of success of 0.2. Calculate the variance for this distribution
1.28 8*0.2(1-0.2)
what shows the possible outcomes of a random experiment and the probability of each outcome?
a probability distribution
continuous random variable
can assume an infinite number of values within a given range. It is measured on a continuous interval or ratio scale. Examples include: The times of commercial flights between Atlanta and Los Angeles are 4.67 hours, 5.13 hours, and so on. The random variable is the time in hours and is measured on a continuous scale of time.
a grocer selects five apples randomly from a box, weighs them and calculates an average weight of 165 grams. match the feature of this process to the correct term
experiment -> selecting and weighing the apples (the experiment is the activity undertaken) outcome -> 165 grams average weight (a specific result of the experiment) event -> an average weight between 150 and 165 grams (an event is combination of one or more outcomes) random variable -> the average weight of five apples (a numerical rep of an outcome)
