4- Probability Distributions and Binomial Distributions
If we roll 2 fair dice, and let x be the sum of the values, what is p(2)?
2/36
If we flip a quarter 4 times, what is the probability of getting exactly 2 tails?
6/16
Determine whether the random variable is discrete or continuous. a. The weight of a T-bone steak. b. The number of fish caught during a fishing tournament. c. The number of bald eagles in the country d. The number of light bulbs that burn out in the next week in a room with 18 bulbs. e. The time it takes to fly from City A to City B.
a.) continuous b.) discrete c.) discrete d.) discrete e.) continuous
Find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p. n=80, p = 0.3
mean= np= 80x0.3=24 variance= npq= 80x0.3x0.7= 16.8 standard deviation= square root of variance= 4.1
Is the expected value of the probability distribution of a random variable always one of the possible values of x? Explain.
no
In a Binomial Distribution, x represents what?
number of successes
E(x2)-(E(x))2 is equal to what?
variance?
What happens as the value of n increases and the probability of success remains the same?
As n increases, the distribution becomes more symmetric.
Decide whether the graph represents a discrete random variable or a continuous random variable. The length of time students use a computer each week
Continuous, because the length of time is a random variable that is uncountable.
What is a discrete probability distribution?
lists each possible value a random variable can assume, together with its probability
In a binomial experiement, what does it mean to say that each trial is independent of the other trials?
Each trial is independent of the other trials if the outcome of one trial does not affect the outcome of any of the other trials.
T/F In most applications, continuous random variables represent counted data, while discrete random variables represent measured data.
False. In most applications, discrete random variables represent counted data, while continuous random variables represent measured data.
T/F The expected value of a random variable can never be negative.
False. The expected value of a random variable can be negative.
In a discrete probability distribution, the sum of the p(x)'s must always be 1.
True
T/F The mean of a random variable represents the "theoretical average" of a probability experiment and sometimes is not a possible outcome.
True
What is the significance of the mean of a probability distribution?
It is the expected value of a discrete random variable.
37% of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each to name his or her favorite nut. Find the probability that the number who say cashews are their favorite nut is (a) exactly three, (b) at least four, and (c) at most two.
P(3)= 0.174 (binompdf(12, .37, 3) P(x_>4)= 1-binomcdf(12, .37, 3)= 0.705 P (x<_2)= binomcdf (12,.37,2)= 0.120
What are the two conditions that determine a probability distribution?
The probability of each value of the discrete random variable is between 0 and 1, inclusive, and the sum of all the probabilities is 1.
Students in a class take a quiz with eight questions. The number x of questions answered correctly can be approximated by the following probability distribution. Complete parts (a) through (e). P(x) 0- 0.02 1- 0.01 2- 0.05 3- 0.06 4- 0.11 5- 0.22 6- 0.31 7- 0.14 8- 0.08 (a) Use the probability distribution to find the mean of the probability distribution. (b) Use the probability distribution to find the variance of the probability distribution. c.) standard deviation d.) find the expected value of the probability distribution.
a.) mean- 5.3 (multiply across) b.) 3.0 c.) 1.7 (square root of 3.0) d.) 5.3 (equal to the mean of the random variable) The expected number of questions answered correctly is 5.3 with a standard deviation of 1.7 questions.
About 20% of babies born with a certain ailment recover fully. A hospital is caring for five babies born with this ailment. The random variable represents the number of babies that recover fully. Decide whether the experiment is a binomial experiment. If it is, identify a success, specify the values of n, p, and q, and list the possible values of the random variable x.
bi nomial experiment? yes what is a success? baby recovers Specify the value of n: n=5 p= 0.20 q= 0.80 List the possible values of the random variable x.--> 0,1,2,3,4,5
A survey asks 1000 workers, "Has the economy forced you to reduce the amount of vacation you plan to take this year?" Twenty-five percent of those surveyed say they are reducing the amount of vacation. Ten workers participating in the survey are randomly selected. The random variable represents the number of workers who are reducing the amount of vacation. Decide whether the experiment is a binomial experiment. If it is, identify a success, specify the values of n, p, and q, and list the possible values of the random variable x.
binomial experiment? yes success? selecting a worker who is reducing the amt of vacation n= 10 p= 0.25 q= 0.75 possible values for x= 0-10
Decide whether the random variable x is discrete or continuous. x represents the amount of snow (in inches) that fell in Nome, Alaska last winter
continuous
Decide whether the random variable x is discrete or continuous. x represents the total number of die rolls required for an individual to roll a five.
discrete