Principle of Counting

Question

Slips of paper marked with the numbers 1, 2, 3, and 4 are placed in a box. A slip is drawn and set aside, its number is recorded, and then a second slip is drawn. Write a sample space with equally likely outcomes for this experiment.

 

Was pretty confused on this one. The wording kept throwing me off.

Answer 1

"Slips of paper marked with the numbers 1, 2, 3, and 4 are placed in a box. A slip is drawn and set aside, its number is recorded, and then a second slip is drawn. Write a sample space with equally likely outcomes for this experiment."

 

First, read and re-read the problem until you understand it and can put it into your own words:

 

"Slips of paper marked with the numbers 1, 2, 3, and 4 are placed in a box."

  A box has 4 slips of paper numbered 1, 2, 3, and 4.

 

"A  slip is drawn and set aside, its number is recorded, and then a second slip is drawn."

  Draw two slips of paper (without replacement) and record the results (in order).

 

"Write a sample space with equally likely outcomes for this experiment."

  List all the possible outcomes that could be recorded.

 

 

The Statistics/math

 

There are 4*3=12 possible outcomes for this experiment (equally-likely).  Enumerated like a counter, they are:

1   2

1   3

1   4

2   1

2   3

2   4

3   1

3   2

3   4

4   1

4   2

4   3

 

Now, when conducting this experiment, you will no longer have to write two numbers for each event (drawing 2 slips of paper), but you can simply count the results in the appropriate row/column on a worksheet:

 

Draw    1:    (1,2) (1,3) (1,4) (2,1) (2,3) (2,4) (3,1) (3,2) (3,4) (4,1) (4,2) (4,3)

Draw    2:                       x

Draw    3:                                                                   x

Draw    4:                                        x

Draw    5:       x

. . .

Draw 100:                                                                                            x

Draw 101:                                                 x

. . .

 

When the total count for each column is calculated, you have the Experimental Probability of these results.  The Theoretical Probability (that is, "equally likely") means that the frequencies are all equal.  As a larger and larger experiment is done, there is a probability that the experimental results with match the theoretical distribution (note: but there remains some small probability that they will not).

Answer 2

I THINK what it means by write a sample space is write all possible outcomes.

 

So if the pieces of paper are marked 1, 2, 3 and 4, you could get 1 then 2, 1 then 3, 1 then 4, 2 then 1...

 

You can write it like this:

(1,2) (1,3) (1,4)

(2,1) (2,3) (2,4)

(3,1) (3,2) (3,4)

(4,1) (4,2) (4,3)

 

Note that (1,1) doesn't work, because once the 1 is drawn, it is not available to be drawn again. The same is true for (2,2), (3,3) and (4,4), so none of these is included in the sample space.

Answer 3

Picture a classroom with 4 columns of chairs facing

the front of the room.  Each column contains 6 chairs,

all of which have a student in them... 24 students in all.

The teacher tells the students to all write the number

of the column in which they are seated on a slip of

paper and drop it into the box she passed around.

 

When all 24 students have done so there are 6 papers

with 1's, 6 papers with 2's, 6 paper's with 3's, and

6 paper's with 4's.

 

Initially, the odds are even for any given number

1-4 to be drawn.  6 out 24 for each number or 1/4.

 

These odds change as each successive slip is drawn, but,

regardless, if she draws all numbers out of the box and

records them she will end up with 6 each of the numbers.

1 to 4.