The student will use theoretical and empirical methods to estimate probabilities.
The student will appraise the differences between the two estimates.
The student will demonstrate an understanding of long-term relative frequencies.
Do the experiment
Count out 40 mixed-color M&Ms® which is approximately one small bag’s worth. Record the number of each color in
[link] . Use the information from this table to complete
[link] . Next, put the M&Ms in a cup. The experiment is to pick two M&Ms, one at a time. Do
not look at them as you pick them. The first time through, replace the first M&M before picking the second one. Record the results in the “With Replacement” column of
[link] . Do this 24 times. The second time through, after picking the first M&M, do
not replace it before picking the second one. Then, pick the second one. Record the results in the “Without Replacement” column section of
[link] . After you record the pick, put
both M&Ms back. Do this a total of 24 times, also. Use the data from
[link] to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Do
not multiply out any fractions.
Population
Color
Quantity
Yellow (
Y )
Green (
G )
Blue (
BL )
Brown (
B )
Orange (
O )
Red (
R )
Theoretical probabilities
With Replacement
Without Replacement
P (2 reds)
P (
R_{1}B_{2} OR
B_{1}R_{2} )
P (
R_{1} AND
G_{2} )
P (
G_{2} |
R_{1} )
P (no yellows)
P (doubles)
P (no doubles)
Note
G_{2} = green on second pick;
R_{1} = red on first pick;
B_{1} = brown on first pick;
B_{2} = brown on second pick; doubles = both picks are the same colour.
Empirical results
With Replacement
Without Replacement
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ )
Empirical probabilities
With Replacement
Without Replacement
P (2 reds)
P (
R_{1}B_{2} OR
B_{1}R_{2} )
P (
R_{1} AND
G_{2} )
P (
G_{2} |
R_{1} )
P (no yellows)
P (doubles)
P (no doubles)
Discussion questions
Why are the “With Replacement” and “Without Replacement” probabilities different?
Convert
P (no yellows) to decimal format for both Theoretical “With Replacement” and for Empirical “With Replacement”. Round to four decimal places.
Theoretical “With Replacement”:
P (no yellows) = _______
Empirical “With Replacement”:
P (no yellows) = _______
Are the decimal values “close”? Did you expect them to be closer together or farther apart? Why?
If you increased the number of times you picked two M&Ms to 240 times, why would empirical probability values change?
Would this change (see part 3) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know?
Explain the differences in what
P (
G_{1} AND
R_{2} ) and
P (
R_{1} |
G_{2} ) represent. Hint: Think about the sample space for each probability.
Questions & Answers
IMAGESNEWSVIDEOS
A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are the mean, median, and mode.
IMAGESNEWSVIDEOS
A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are th
Ahmed
what is confidence interval estimate and its formula in getting it
There are two coins on a table. When both are flipped, one coin land on heads eith probability 0.5 while the other lands on head with probability 0.6. A coin is randomly selected from the table and flipped.
(a) what is probability it lands on heads?
(b) given that it lands on tail, what is the Condi
outlier is an observation point that is distant from other observations.
Gidigah
what is its effect on mode?
Usama
Outlier have little effect on the mode of a given set of data.
Gidigah
How can you identify a possible outlier(s) in a data set.
Daniel
The best visualisation method to identify the outlier is box and wisker method or boxplot diagram. The points which are located outside the max edge of wisker(both side) are considered as outlier.
Akash
@Daniel Adunkwah - Usually you can identify an outlier visually. They lie outside the observed pattern of the other data points, thus they're called outliers.
Ron
what is completeness?
Muhammad
I am new to this. I am trying to learn.
Dom
I am also new Dom, welcome!
Nthabi
thanks
Dom
please my friend i want same general points about statistics. say same thing
sorry question a bit unclear...do you mean how do you analyze quantitative data? If yes, it depends on the specific question(s) you set in the beginning as well as on the data you collected. So the method of data analysis will be dependent on the data collecter and questions asked.
Bheka
how to solve for degree of freedom
saliou
Quantitative data is the data in numeric form. For eg: Income of persons asked is 10,000. This data is quantitative data on the other hand data collected for either make or female is qualitative data.
Rohan
*male
Rohan
Degree of freedom is the unconditionality. For example if you have total number of observations n, and you have to calculate variance, obviously you will need mean for that. Here mean is a condition, without which you cannot calculate variance. Therefore degree of freedom for variance will be n-1.
Rohan
data that is best presented in categories like haircolor, food taste (good, bad, fair, terrible) constitutes qualitative data
Bheka
vegetation types (grasslands, forests etc) qualitative data
Bheka
I don't understand how you solved it can you teach me