Question

Suppose you toss a fair coin 4 times. Denote the outcome to be 1 if you get a head and 0 if a tail.

a) Write down the sample space Ω

b) What is the probability of the event that you get head at least once?

c) If you get four same toss you will get 10 dollars, otherwise you will lose 2 dollars. On average, will you win or lose?

Answer 1

(a)

Sample Space = {1,1,1,1), (0,1,1,1), (1,1,1,0),(0,1,1,0),(1,1,0,1),(0,1,0,1),(1,1,0,0),(0,1,0,0),(1,0,1,1),(0,0,1,1),(1,0,1,0),(0,0,1,0),(1,0,0,1),(0,0,0,1),(1,0,0,0),(0,0,0,0)} : 16 Nos.

(b)

Favorable events of the event that you get head at least once: {1,1,1,1), (0,1,1,1), (1,1,1,0),(0,1,1,0),(1,1,0,1),(0,1,0,1),(1,1,0,0),(0,1,0,0),(1,0,1,1),(0,0,1,1),(1,0,1,0),(0,0,1,0),(1,0,0,1),(0,0,0,1),(1,0,0,0): 15 Nos.

So,

the probability of the event that you get head at least once = 15/16 = 0.9375

So,

Answer is:

0.9375

(c)

Favorable events of the event : four same toss : (1,1,1,1), (0,0,0,0): 2 Nos.

p = the probability of the event: four same toss ; 2/16 = 0.0769

So,

q = 1 - p = 0.9231

From the given data the following Table is calculated:

Event	Value (x)	Probability (p)	x p
you get four same toss	10	0.0769	10 X 0.0769 = 0.769
you do not get four same toss	- 2	0.9231	- 2 X 0.9231 = - 1.8462
Expected value			- 1.0772

On average, you will lose.

Loss: - $1.08