Question

A coin is tossed 5 times. X is the number of heads observed. Find the probability distribution of X.

Solution

Here X denotes the number of heads obtained in five tosses of a coin. X can take the values 0, 1, 2, 3, 4, 5.
Let p be the probability of getting a head and q be the probability of not getting a head.
$therefore space space space straight p space equals space 1 half comma space space space straight q space equals space 1 minus straight p space equals space 1 minus 1 half space equals 1 half$
$straight P left parenthesis straight X space equals space 0 right parenthesis space equals straight q space straight q space straight q space straight q space straight q space equals space 1 half cross times 1 half cross times 1 half cross times 1 half cross times 1 half equals space 1 over 32 straight P left parenthesis straight X space equals space 1 right parenthesis space equals space straight p space straight q space straight q space straight q space straight q space plus space straight q space straight p space straight q space straight q space straight q space space plus space straight q space straight q space straight p space straight q space straight q space space plus space straight q space straight q space straight q space straight p space straight q space plus space straight q space straight q space straight q space straight q space straight p space space space space space space space space space space space space space space space space space equals space 1 over 32 plus 1 over 32 plus 1 over 32 plus 1 over 32 plus 1 over 32 space equals 5 over 32 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight p space straight p space straight q space straight q space straight q space space plus space space.......... space plus space straight q space straight q space straight q space straight p space straight p space space space space space space space space space space space space space space space space space space space equals space 1 over 32 plus 1 over 32 plus..... plus 1 over 32 space equals space 10 over 32 straight P left parenthesis straight X space equals space 3 right parenthesis space equals space straight p space straight p space space straight p space straight q space straight q space plus............ plus straight q space straight q space straight p space straight p space straight p space space space space space space space space space space space space space space space space space space space space equals space 1 over 32 plus 1 over 32 plus.... plus 1 over 32 space equals space 10 over 32 straight P left parenthesis straight X space equals space 4 right parenthesis space equals space straight p space straight p space straight p space straight p space straight q space plus space. space... space plus space straight q space straight p space straight p space straight p space straight p space equals space 1 over 32 plus 1 over 32 plus 1 over 32 plus 1 over 32 plus 1 over 32 space equals 5 over 32 straight P left parenthesis straight X space equals space 5 right parenthesis space equals space straight p space straight p space straight p space straight p space straight p space equals space 1 half cross times 1 half cross times 1 half cross times 1 half cross times 1 half cross times 1 half space equals 1 over 32$
∴ probability distribution of X is

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Probability

A coin is tossed 5 times. X is the number of heads observed. Find the probability distribution of X.

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