Step 1: Determine how many people are there..
There are \(\displaystyle 11\) people.
Step 2: Determine how many people shake hands in a handshake...
\(\displaystyle 2\) people make one handshake.
Step 3: Determine how many handshakes can be made...
We have a restriction here, so we need to use permutation..

So, there will be \(\displaystyle 11P2\) handshakes.

\(\displaystyle 11P2=\frac {11*10}{2*1}=11*5=55\) handshakes

Step 1: Count how many letters are in the word ANACONDA...

There are 8 letters.

Step 2: Count how many repeats of any letters (if any)..

There are \(\displaystyle 3\) A's and \(\displaystyle 2\) N's.

Step 3: Find how many ways I can rearrange...

\(\displaystyle \frac {8!}{3!*2!}=\frac {8*7*6*5*4*3*2*1}{3*2*1*2*1}...\)

\(\displaystyle ...=\frac {8*7*6*5*4}{2}=8*7*6*5*2=3360\)

Step 1: Count how many numbers are in the word...

There are \(\displaystyle 12\) letters. 

Step 2: Count the number of repeated letters...

There are \(\displaystyle 2\) T's

There are \(\displaystyle 2\) O's

There are \(\displaystyle 2\) R's

Step 3: To find how many ways I can arrange the letters, take the factorial of the total number of letters and divide it by the factorial of how many times a certain letter repeats...

So, \(\displaystyle \frac {12!}{2!\times2!\times2!}\)

Because x is a continuous uniform random variable the expected value and variance can be found with the following formulas:

\(\displaystyle E[x]=\frac{a+b}{2}\)

\(\displaystyle Var(x)=\frac{(b-a)^2}{12}\)

X is uniform on (a,b). In this case a is 0 and b is 50. Plugging the values of a and b into the given formulas will give the answers:

\(\displaystyle E[x]=\frac{a+b}{2}=\frac{0+50}{2}=25\)

\(\displaystyle Var(x)=\frac{(b-a)^2}{12}=\frac{2500}{12}=208.33\)

This problem uses the Binomial Distribution: \(\displaystyle \binom{n}{k}p^k(1-p)^{n-k}\)

For this problem n is the number of trials, or 15. Because the problem stated that the coin was a fair coin the probability of heads is one half, or .5.

The binomial distribution is a discrete distribution so the expression x<3 has to be broken down.

\(\displaystyle Pr(x< 3)=Pr(x=0)+Pr(x=1)+Pr(x=2)\)

\(\displaystyle Pr(x=0)=\begin{pmatrix} 15\\0 \end{pmatrix} (.5)^0(1-.5)^{15}=.000031\)

\(\displaystyle Pr(x=1)=\begin{pmatrix} 15\\1 \end{pmatrix} (.5)^1(1-.5)^{14}=.00046\)

\(\displaystyle Pr(x=2)=\begin{pmatrix} 15\\2 \end{pmatrix} (.5)^2(1-.5)^{13}=.0032\)

Adding the probabilities will give the final answer.

\(\displaystyle .000031+.0046+.0032=.0037\)

The first step in this problem is calculating the z-score. \(\displaystyle z=( x-\mu)/\sigma\)

\(\displaystyle (95-85)/2.5= 4\)

The next step is to look up 4 in the z-table. The value from the table is \(\displaystyle .0000317\).

\(\displaystyle 1-.0000317\: is \: .9999683.\)

A \(\displaystyle z\)-score indicates whether a particular value is typical for a population or data set.  The closer the \(\displaystyle z\)-score is to 0, the closer the value is to the mean of the population and the more typical it is.  The \(\displaystyle z\)-score is calculated by subtracting the mean of a population from the particular value in question, then dividing the result by the population's standard deviation. 

\(\displaystyle z=\frac{X-\bar{X}}S{}\) 

\(\displaystyle (115-103)/8=1.5\)

A Z-score indicates whether a particular value is typical for a population or data set.  The closer the Z-score is to 0, the closer the value is to the mean of the population and the more typical it is.  The Z-score is calculated by subtracting the mean of a population from the particular value in question, then dividing the result by the population's standard deviation.

To find the z-score, follow the formula

\(\displaystyle z=\frac{x-\mu}{\sigma }, x=score, \mu=mean, \sigma=S.D.\)

\(\displaystyle z=\frac{73-55}{5}\)

or

\(\displaystyle z=3.6\)

A z score is unique to each value within a population.

To find a z score, subtract the mean of a population from the particular value in question, then divide the result by the population's standard deviation.

\(\displaystyle z=\frac{x-\mu}{\sigma}=\frac{19.5-17.2}{3.5}= \frac{2.3}{3.5}=0.6571\)