When a fair coin is tossed 4 times then the sample space is
S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT}
∴ n(S) = 16
X denotes the number of heads.
∴ X can take the value 0, 1, 2, 3, 4
When X = 0, then X = {TTTT}
∴ n (X) = 1
∴ P(X = 0) = \(\frac{n(X)}{n(S)}\) = \(\frac{1}{16}\) = \(\frac{^4C_0}{16}\)
When X = 1, then
X = {HTTT, THTT, TTHT, TTTH}
∴ n(X) = 4
∴ P(X = 1) = \(\frac{n(X)}{n(S)}\) = \(\frac{4}{16}\) = \(\frac{^4C_1}{16}\)
When X = 2, then
X = {HHTT, HTHT, HTTH, THHT, THTH, TTHH}
∴ n(X) = 6
∴ P(X = 2) = \(\frac{n(X)}{n(S)}\) = \(\frac{6}{16}\) = \(\frac{^4C_2}{16}\)
When X = 3, then
X = {HHHT, HHTH, HTHH, THHH}
∴ n(X) = 4
∴ P(X = 3) = \(\frac{n(X)}{n(S)}\) = \(\frac{4}{16}\) = \(\frac{^4C_3}{16}\)
When X = 4, then X = {HHHH}
∴ n(X) = 1
∴ P(X = 4) = \(\frac{n(X)}{n(S)}\) = \(\frac{1}{16}\) = \(\frac{^4C_4}{16}\)
∴ the probability distribution of X is as follows :
| x |
0 |
1 |
2 |
3 |
4 |
| p(x) |
1/16 |
4/16 |
6/16 |
4/16 |
1/16 |
Also, the formula for p.m.f. of X is
P(x) = \(\frac{^4C_x}{16}\), x = 0, 1, 2, 3, 4 and = 0, otherwise.
