Q. 65.0( 2 Votes )

# Find the mean, va

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 p_{i} and x_{i.}

Let’s proceed to find mean and variance.

Mean of any probability distribution is given by Mean = ∑x_{i}p_{i}

Variance is given by:

Variance = ∑ x_{i}^{2}p_{i} – (∑x_{i}p_{i})^{2}

Standard Deviation is given by SD = √ Variance

∴ first we need to find the products i.e. p_{i}x_{i} and p_{i}x_{i}^{2} 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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