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A person plays a game of tossing a coin thrice. For each tail, he is given Rs.3 by the organiser of the game and for each head, he has to give Rs.2 to the organiser. Let X denote the amount gained or lost by the person. Show that X is a random variable and exhibit it as a function on the sample space of the experiment.

Text Solution

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`X` is a number whose values are defined on the outcomes of a random experiment. Therefore `X` is a random variable.
Now, sample space of the experiment is
`S = {"HHH, HHT, HTH, THH, HTT, THT, TTH, TTT"}`
Then `"X(TTT)" `= `Rs.(3xx3)=Rs.9`
`"X(HHH) = X(HTH)=X(THH) "`
`=Rs.(3xx1-2xx2)`
` = Rs. (-1)`
`"X(HTT)=X(THT)=X(TTH) "`
`=Rs.(3xx2-2xx1)`
`= Rs.(4) `
`"X(HHH)" = - Rs.(3xx2)`
`= - Rs.6`.
Where, minus sign shows the loss to the player. Thus for each element of the sample space. `X` takes a unique value. Hence `X` is a function on the sample space whose range is
`{-6, -1, 4, 6}`
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