A fair coin is flipped three times. Let X be the number of heads observed. (a) Give both the possible values and probability mass function of X. (b) Find P(X ≥ 1) and P(X > 1). (c) Compute E[X] and V ar(X).

a)

The possible values are 0, 1, 2 and 3.

The probability mass function of X is:

P(X = 0) = 0.125

P(X = 1) = 0.375

P(X = 2) = 0.375

P(X = 3) = 0.125

b)

,

c) E(X) = 1.5, Var(X) = 0.75

Step-by-step explanation:

For it time the coin is flipped, there are only two possible outcomes. Either it comes up heads, or it comes up tails. Each toss is independent of other tosses. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

In which is the number of different combinations of x objects from a set of n elements, given by the following formula.

And p is the probability of X happening.

The expected value of the binomial distribution is:

The variance of the binomial distribution is:

A fair coin is flipped three times.

This means that

Fair coin means that it is equally as likely to be heads or tails, so

a) Give both the possible values and probability mass function of X.

The possible values are from 0 to n, so 0, 1, 2 and 3.

The probability mass function is the probability of each outcome. So

(b) Find P(X ≥ 1) and P(X > 1).

(c) Compute E[X] and Var(X).

The answer is in the image 

Salesperson will make 6% of 1800

=(6/100)*1800

=108

Salesperson will make $108 in $1800 sales

The solution is given in the image below

The wood before starting =12 feet

Left wood=6 feet

Wood used till now=12-6=6 feet

Picture frame built till now= 6/(3/4)

=8 pieces

Therefore, till now 8 pieces have been made.

The wood before starting =12 feet

Left wood=6 feet

Wood used till now=12-6=6 feet

Picture frame built till now= 6/(3/4)

=8 pieces

Therefore, till now 8 pieces have been made.

The wood before starting =12 feet

Left wood=6 feet

Wood used till now=12-6=6 feet

Picture frame built till now= 6/(3/4)

=8 pieces

Therefore, till now 8 pieces have been made.

Let the father's age be x and son's be y

10 years before-

Father age=x-10

sons age=y-10

Given,

x-10=10(y-10)

x-10=10y-100

Given present age of father=40

therefore,

x=40

40-10=10y-100

10y-100=30

10y=130

y=130/10

y=13

Therefore present age of son=13years

Volume of rectangular prism=length * width * height

Given,

length=3/8cm

breadth=5/8cm

height=7/8cm

therefore volume=(3/8)*(5/8)*(7/8)

                           =105/512 cm3

Cost before sale tax=$46.00

Cost after including tax=$48.76

Sale tax=$48.76-46.00

=$2.76

we know,

sale tax=((sales tax rate)/100) * cost before sale tax

2.76=((sales tax rate)/100) *46.00

sale tax rate=(2.76*100)/46

                   =276/46

                    =6%

There fore sales tax rate=6%