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 some heads in 3 tosses of a coin.

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

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

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

Thus,

P(X = 0) = P(TTT) = P(T)P(T)P(T) = 1/2 x 1/2 x 1/2 = 1/8

P(X = 1) = 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 = 2) = 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 = 3) = P(HHH) = P(H)P(H)P(H) = 1/2 x 1/2 x 1/2 = 1/8

Now we have p_i and x_i.

As

Thus, ∑(p_i) = 1

∴ Now we are ready to write the probability distribution for X:-

The following table gives probability distribution: