Question

Toss 5 coins 25 times and note on each throw the number of heads. Make a probability distribution of the number of heads. Find mean and variance of that distribution and compare it with the mean and variance of theoretical probability distribution using binomial probability distribution.

Answer 1

Toss	Coin 1	Coin 2	Coin 3	Coin 4	Coin 5	# of Heads
1	1	1	0	0	0	3
2	1	0	0	1	1	2
3	1	0	1	0	0	3
4	1	0	1	1	1	1
5	1	0	0	1	0	3
6	1	1	0	1	0	2
7	1	0	1	1	0	2
8	1	0	0	1	1	2
9	0	0	0	1	0	4
10	1	1	1	0	0	2
11	1	1	0	1	0	2
12	0	1	0	0	0	4
13	1	0	1	1	0	2
14	1	0	0	0	0	4
15	1	0	0	0	0	4
16	0	0	1	1	1	2
17	0	1	0	0	0	4
18	0	1	1	0	0	3
19	1	1	1	0	1	1
20	1	0	0	1	1	2
21	0	1	0	0	1	3
22	0	0	0	0	0	5
23	0	0	0	1	0	4
24	0	1	0	1	1	2
25	0	0	0	1	0	4

# of Heads (x)	Frequency	Probability P(x)
0	0	0
1	2	0.08
2	10	0.4
3	5	0.2
4	7	0.28
5	1	0.04

Mean = ∑x * P(x) = 2.8

Variance = ∑(x^2) P(x) - Mean^2 = 1.12

For the theoretical binomial distribution, mean = np = 5 * (1/2) = 2.5 and variance = np(1 - p) = 5 * (1/2) * (1/2) = 1.25