. Problem
Find the probability distribution of
(i) number of heads in two tosses of a coin. (ii) number of tails in the simultaneous tosses of three coins.
(iii) number of heads in four tosses of a coin.
II. Solution
Let Y denote the event of tossing a coin. Considering a fair coin, the probability of getting a Head
or Tail P(Y) = 0.5
We have i coin tosses, with probability p of Heads and (1-p) of Tails. We conduct the trials independently.
In general , the probability of getting of j Head/Tail is given as:
n! j (n−j)
P(Y = j) = p (1 − p) (1) j!(n − j)!
Consider a random variable X where X = Number of successes.Suppose we have n trials. We write X
˜B(n, p)
(i) Let X denote the random variable of number of Heads. The probability distribution of getting exactly j Heads in 2 tosses of coin is given as:
X ˜B(2, 0.5)
Using equation
2! .5j(1 − 0.5)(2−j) (2)
P(X = j) = 0 j!(2 − j)!
We get the pdf as below:
P(X = 0) = .50(1 − 0.5)(2−0) = 0.25 (3)
P(X = 1) = 0.51(1 − 0.5)(2−1) = 0.5 (4)
P(X = 2) = .52(1 − 0.5)(2−2) = 0.25 (5)
The distribution table is given as:
j
0
1
2
P(X=j)
0.25
0.5
0.25
(ii)Let X denote the random variable of number of Tails. The probability distribution of getting exactly j Tails in 3 tosses of coin is given as:
X ˜B(3, 0.5)
Using equation
3! .5j(1 − 0.5)(3−j) (6)
P(X = j) = 0 j!(3 − j)!
We get the pdf as below:
P(X = 0) = 0.50(1 − 0.5)(3−0) = 0.125
(7)
P(X = 1) = 0.51(1 − 0.5)(3−1) = 0.375
(8)
P(X = 2) = 0.52(1 − 0.5)(3−2) = 0.375
(9)
P(X = 3) = 0.53(1 − 0.5)(3−3) = 0.125
(10)
The probability distribution of X is:
j
0
1
2
3
P(X=j)
0.125
0.375
0.375
0.125
(iii)Let X denote the random variable of number of Heads. The probability distribution of getting exactly j Heads in 4 tosses of coin is given as:
X ˜B(4, 0.5)
Using equation
4! j (4−j)
P(X = j) = 0.5 (1 − 0.5) (11) j!(4 − j)!
2
We get the pdf as below:
P(Y3 = 0) = 0.50(1 − 0.5)(4−0) = 0.0625
(12)
P(X = 1) = 0.51(1 − 0.5)(4−1) = 0.25
(13)
P(X = 2) = 0.52(1 − 0.5)(4−2) = 0.375
(14)
P(X = 3) = 0.53(1 − 0.5)(4−4) = 0.25
(15)
P(X = 4) = 0.54(1 − 0.5)(4−4) = 0.0625
(16)
The probability distribution of X is:
j
0
1
2
3
4
P(X=j)
0.0625
0.25
0.375
0.25
0.0625
The probabilities were simulated using the python code.
Figure 1: Simulation for tossing a fair coin
