.In a group of 1,000 persons, there are 650 who can speak Hindi, 400 can speak English and 150 can speak both Hindi and English. If a person is selected at random, what is the probability that he speaks

Hindi only,

English only,

only one of the two languages

at least one of the two languages?5 marks

2. A bag contains 2 red and 1 black ball and another bag contains 2 red and 2 black balls. One ball is selected at random from each bag. Find the probability of drawing

(a) at least a red ball,

(b) a black ball from the second bag given that ball from the first is red;

(c) show that the event of drawing a red ball from the first bag and the event of drawing a red ball from the second bag are independent.5 marks

3. In a binomial distribution consisting of 5 independent trials, the probability of 1 and 2 successes are 0.4096 and 0.2048 respectively. Calculate the mean, variance and mode of the distribution. 5marks

4. A manager accepts the work submitted by his typist only when there is no mistake in the work. The typist has to type on an average 20 letters per day of about 200 words each. Find the chance of her making a mistake

(i) if less than 1% of the letters submitted by her are rejected;

(ii) if on 90% of days all the work submitted by her is accepted. [As the probability of making a mistake is small, you may use Poisson distribution.5 marks

5. The distribution of 1,000 examinees according to marks percentage is given below :

% Marksless than 4040 - 7575 or moreTotal

No. of examinees4304201501000

Assuming the marks percentage to follow a normal distribution, calculate the mean and standard deviation of marks. If not more than 300 examinees are to fail, what should be the passing marks?5 marks

6.Marks in an examination are approximately normally distributed withv mean 75 and standard deviation 5. If the top 5% of the students get grade A and the bottom 25% get grade F, what mark is the lowest A and what mark is the highest F?5 Marks