A fair coin is tossed 4 times. Let X denote the number of heads obtained. Write down the probability distribution of X.

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A fair coin is tossed 4 times. Let X denote the number of heads obtained. Write down the probability distribution of X. Also, find the formula for p.m.f. of X.


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Salim Ahmed Kawsar

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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.

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