SOLUTION: Let random variable x represent the number of heads when a fair coin is tossed three times. (a) Construct a table describing the probability distribution. (b) Deter

Before making the table we have to do some other things first.

List the sample space, which is the list of all ways the coins 
can land:

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

Each of those are equally likely and since there are 8 of them
each has probability 1/8, which is 0.125 as a decimal.

There is only 1 case with 0 heads, TTT, which has probability 1/8.
That's the case when X=0 and P(X)=P(0)=1/8=0.125

There are 3 cases with 1 heads, HTT, THT, TTH, So the probability
of 1 head is 3 times 1/8 or 3/8 or 0.375 That's the case when X=1 
and P(X)=P(1)=3/8=0.375.

There are 3 cases with 2 heads, HHT, HTH, HHT, So the probability
of 2 heads is also 3 times 1/8 or 3/8 or 0.375 That's the case 
when X=2 and P(X)=P(2)=3/8=0.375.

There is only 1 case with 3 heads, HHH, which has probability 1/8.
That's the case when X=3 and P(X)=P(3)=1/8=0.125.

Now we can make the table:

 X   P(X)
 --------
 0   0.125
 1   0.375
 2   0.375
 3   0.125


That's the answer to (a).

We make a new table, starting with the same first column.

  X
  -----------------
  0
  1
  2
  3
  ----------------

Under the bottom line we determine the mean by averaging the 4
numbers of heads in all the rolls. We add the column and divide
by 4:

  X
  -----------------
  0
  1
  2
  3
  ----------------  
4)6.0
  1.5

the mean = m = (0+1+2+3)÷4 = 6÷4 = 6/4 = 3/2 = 1.5
So the mean = m = 1.5. Next we subtract the mean of 1.5 from
each of the values of X:
For X=0, we subtract 1.5 from 0 and get X-m=0-1.5 = -1.5
For X=1, we subtract 1.5 from 1 and get X-m=1-1.5 = -0.5
For X=2, we subtract 1.5 from 2 and get X-m=2-1.5 = +0.5
For X=3, we subtract 1.5 from 3 and get X-m=3-1.5 = +1.5

Then we list those in a new column out beside the first:

  X      X-m
  -----------------
  0     -1.5
  1     -0.5
  2     +0.5
  3     +1.5
  ----------------  
4)6.0         
  1.5

Next we square each of the values of X-m:
For X-m = -1.5, we square and get (X-m)²=(-1.5)²= 2.25
For X-m = -0.5, we square and get (X-m)²=(-0.5)²= 0.25
For X-m = +0.5, we square and get (X-m)²=(+0.5)²= 0.25
For X-m = +1.5, we square and get (X-m)²=(+1.5)²= 2.25
Then we list those in a new column out beside the second:

  X      X-m     (X-m)²
  ---------------------
  0     -1.5     2.25  
  1     -0.5     0.25
  2     +0.5     0.25
  3     +1.5     2.25
  -------------------  
4)6.0                 
  1.5

Under the bottom line we determine the variance by averaging the 4
values of (X-m)² in all the rolls. We add the column and divide
by 4.

NOTICE: We divide by 4 because this is a POPULATION, not a SAMPLE.
If it were a SAMPLE, we would divide by 1 less than 4, or 3.  But
this is the ENTIRE POPULATION of ALL the ways the coin can land on
the three tosses, so we divide by 4, not 3:

  X      X-m     (X-m)²
  ---------------------
  0     -1.5     2.25  
  1     -0.5     0.25
  2     +0.5     0.25
  3     +1.5     2.25
  -------------------  
4)6.0          4)5.00         
  1.5=m          1.25=s²=the variance

Finally we find the standard deviation, s, by taking the
square root of the variance s². The square root of 1.25
is approximately 1.118033989, so that's the standard 
deviation s.  Round that off 
as your teacher told you.

  X      X-m     (X-m)²
  ---------------------
  0     -1.5     2.25  
  1     -0.5     0.25
  2     +0.5     0.25
  3     +1.5     2.25
  -------------------  
4)6.0          4)5.00        
  1.5=m          1.25=s²=the variance
                 1.118033989=s=the standard deviation

Edwin