Answer :

When we toss a coin three times we have the following possibilities:


{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}


Let X be a random variable representing number of tails in 3 tosses of a coin.


probability of getting a head or probability of getting a tail are independent events and P(GETTING A HEAD) = P(GETTING A TAIL) = 1/2


P(Head in first toss) and P(Head in second toss) and P(head in third toss) can be given by their individual products.


Note: P(AՈB) = P(A)P(B) where A and B are independent events.


Thus,


P(X=0) = P(HHH) = P(H)P(H)P(H) = 1/2 x 1/2 x 1/2 = 1/8


P(X=1) = P(HHT or HTH or THH) = P(HHT)+P(HTH)+P(THH)


= P(H)P(H)P(T)+ P(H)P(T)P(H)+ P(T)P(H)P(H)


= 1/2 x 1/2 x 1/2 + 1/2 x 1/2 x 1/2 + 1/2 x 1/2 x 1/2


= 3/8


P(X=2) = P(HTT or THT or TTH) = P(HTT)+P(THT)+P(TTH)


= P(H)P(T)P(T)+ P(T)P(H)P(T)+ P(T)P(T)P(H)


= 1/2 x 1/2 x 1/2 + 1/2 x 1/2 x 1/2 + 1/2 x 1/2 x 1/2


= 3/8


P(X=3) = P(TTT) = P(T)P(T)P(T) = 1/2 x 1/2 x 1/2 = 1/8


Now we have pi and xi.


Let’s proceed to find mean and variance.


Mean of any probability distribution is given by Mean = ∑xipi


Variance is given by:


Variance = ∑ xi2pi – (∑xipi)2


Standard Deviation is given by SD = √ Variance


first we need to find the products i.e. pixi and pixi2 and add them to get mean and apply the above formula to get the variance.


Following table gives the required products :



Mean =


Variance =


Standard Deviation = √(3/4) = = 0.87


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