Question

Three fair coins are flipped independently. Let X be the number of heads among the three coins.

(1) Write down all possible values that X can take.

(2) Construct the probability mass function of X.

(3) What is the probability that we observe two or more heads. (i.e., P(X ≥ 2))

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

Answer 1

Solution-:

(1) Three fair coins are flipped independently. Let X be the number of heads among the three coins.

The 8 possible elementary events, and the corresponding values for X, are:

Elementary Event	Value of X
HHH	3
HHT	2
HTH	2
THH	2
HTT	1
THT	1
TTH	1
TTT	0

(2) Therefore, the probability distribution for the number of heads occurring in three coin tosses is:

X	P(x)
0	1/8=0.125
1	3/8=0.375
2	3/8=0.375
3	1/8=0.125
Total	1

(3) we find, P[ the probability that we observe two or more heads]

  

Therefoe, the required probability is 0.50

(4)

X	P(x)	X*P(x)	X^2*P(x)
0	0.125	0	0
1	0.375	0.375	0.375
2	0.375	0.75	1.5
3	0.125	0.375	1.125
Total	1	1.50	3.0

Mean=

and

Variance=