Question

(i) State and prove the theorem of total probability. 

(ii) If a fair coin is tossed 10 times, what is the probability that the outcome is exactly 6 heads?

Answer 1

Let {E₁ ,E₂ ,….,E₃}be a partition of the sample space S, and suppose that each of the events E₁ , E₂ ,….,E₃  as nonzero probability of occurrence. Let A be any event associated with S, then

P(A) = P(E₁) P(A/E₁) + P(E₂) P(A/E₂) +...+P(E_n) P(A/E_n)}

Proof:

By multiplication rule of probability we have;

\(P(A)=P\left(E_{1}\right) P\left(A / E_{1}\right)+P\left(E_{2}\right) P\left(A / E_{2}\right)+\ldots . .+P\left(E_{n}\right) P\left(A / E_{n}\right)\)

(ii) Let X denotes the random variable of number of heads in an experiment of 10 trials. Clearly X has a Binomial Distribution with n = 10

 Here n = 10, p = 1/2, q = 1 – p = 1/2