Suppose you have an experiment where you flip a coin three times. You then count the number of heads.

Question

a.)    State the random variable. X = number of heads observed when coin is flipped 3 times

b.)   Write the probability distribution for the number of heads.

c.)    Draw a histogram for the number of heads.

d.)   Find the mean number of heads.

e.)    Find the variance for the number of heads.

f.)     Find the standard deviation for the number of heads.

g.)    Find the probability of having two or more number of heads.

h.)   Is it unusual to flip two heads?

I got the random variable, but am so stuck after that. Please help.

Answer 1

a) Then X =0,1,2,3

b)

Prob(X=0) = 1/2 * 1/2*1/2 = 1/8

the head can appear in 1 of 3 positions

Prob(X=1) = 1/2 * 1/2*1/2 *3 = 3/8

the tail can appear in 1 of 3 positions

Prob(X=2) = 1/2*1/2*1/2 *3 = 3/8

Prob(X=3) = 1/2*1/2*1/2 = 1/8

c)

0 *

1 ***

2 ***

3 *

d)

expected value

= SUM OF [ outcome * prob]

= 0*1/8 + 1*3/8 + 2*3/8 + 3*1/8

= 3/8+6/8+3/8

= 12/8

= 3/2

e)

Var = Expected Value of x^2 - (Expected value of x)^2

Expected value of x^2 =

= 0^2 * 1/8 + 1^2 * 3/8 + 2^2 * 3/8 + 3^2*1/8

= 1*3/8 + 4*3/8 + 9*1/8

= 3/8 + 12/8 + 9/8

= 24/8

= 3

variance = 3 - 3/2 ^2

= 3 - 9/4

= 12/4 - 9/4

= 3/4

f) standard deviation is the square root of the variance

sqrt(3/4) = sqrt(3)/2

g) Prob(X>=2) = 3/8+1/8 = 4/8 = 1/2 = 50%

so h) No it is likely to happen