Find the mean number of heads in three tosses of a fair coin.

Solution

Here X denotes the number of heads obtained in three tosses of a coin. Thus X can take the values 0, 1, 2, 3.
Let p be the probabilities of getting a head and q be the probability of not getting a head,
$therefore space space space space straight p space equals space 1 half comma space space straight q space equals space 1 space minus space straight p space equals space 1 space space minus 1 half space equals space 1 half$
$straight P left parenthesis straight X space equals space 0 right parenthesis space equals space straight q space cross times space straight q space cross times space straight q space space equals space 1 half cross times 1 half cross times 1 half space equals 1 over 8 straight P left parenthesis straight X space equals space 1 right parenthesis space equals space straight p space straight q space straight q space space plus space straight q space straight p space straight q space plus space straight q space space straight q space straight p space space space space space space space space space space space space space space space equals space 1 half cross times 1 half cross times 1 half plus 1 half cross times 1 half cross times 1 half plus 1 half cross times 1 half cross times 1 half space equals 1 over 8 plus 1 over 8 plus 1 over 8 space equals 3 over 8 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight p space straight p space straight q space plus space straight p space straight q space straight p space plus space straight q space straight p space straight p space space space space space space space space space space space space space space space equals space 1 half cross times 1 half cross times 1 half plus 1 half cross times 1 half cross times 1 half plus 1 half cross times 1 half cross times 1 half space equals 1 over 8 plus 1 over 8 plus 1 over 8 space equals 3 over 8 straight P left parenthesis straight X space equals space 3 right parenthesis space equals space straight p space straight p space straight p space equals space 1 half cross times 1 half cross times 1 half space equals 1 over 8$
∴ probability distribution table is

$therefore space space space straight mu space equals space sum from blank to blank of space straight x space straight p subscript straight i space equals space 3 over 8 plus 6 over 8 plus 3 over 8 space equals 12 over 8 space equals 3 over 2 space equals space 1.5$

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Probability

Find the mean number of heads in three tosses of a fair coin.

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