Example on Probability

                                       Mathematical Statistics.

                                                Probability

Q1. The probability that a student will pass the final examination in both English and

       Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of

       passing the English examination is 0.75, what is the probability of passing the

       Hindi examination?

                                                                 SOLUTION.

Let E and F denote the events of passing the English and Hindi Examination respectively

Then, P(E) = 0.75 and P(F) = x (to be calculated)  and   P(E ∩ F) = 0.5 and P(E´∩ F´) = 0.1  … given

Now, we know that,

Also E´∩ F´ = (E ∪ F)´               (by Demorgan's Law)

Therefore, P(E∪F)´ = P(E´∩ F´) = 0.1    …given

Now,since,

                         P(E∪F) + P(E ∪ F)´ = 1

Therefore,

                  P(E ∪ F) = 1 - P(E ∪ F)´ = 1- 0.1 = 0.9

Now , P(E ∪ F) = P(E) + P(F) – P(E ∩ F)………well known axiom of probability

or P(E ∪ F) = 0.75 + x – 0.5

or, 0.9 = 0.75 + x – 0.5

• X = 0.9 – 0.25 = 0.65

Therefore P( F) = 0.65

            Therefore,the probability of passing the Hindi examination is 0.65