Write down the sample spaces for the following experiments:

i) A coin is tossed and at the same time a die is rolled.

ii) The order in which a mouse, a frog, and a rabbit arrive at a lake is observed.

Find the probability of drawing an ace or a spade from a deck of 52 cards in a single draw.

From 7 men and 4 women a committee of 6 is to be formed. In how many ways can this be done (i) when the committee contains exactly 2 women, (ii) at least 2 women?

The following data were obtained in a study of the relationship between the resistance (ohms) and the failure time (minutes) of certain overloaded resistors.

Which of the following statements are true? Give reasons for your answers.

i) The derivative of the quotient of two functions is equal to the quotient of the derivatives of the two functions.

ii) The function of f ( x) = sin x/2 + sinx has a maxima at x = π/2 .

iii) 2x + 3 y + 4z = 0 represents a line in 3-dimenisonal space.

v) In a Binomial distribution if mean is known then standard deviation is also known.

v) If two events are mutually exclusive then they are independent.

Four groups of students were subjected to different teaching techniques and tested at the end of a specified period of time. As a result of dropouts from the experimental groups the number of students varied from group to group. Do the data shown below present sufficient

evidence to indicate a difference in the mean achievement for the four teaching techniques at 0.01 level of significance? What is the conclusion if you fix the level of significance at 0.05?

Verify that f (x) = x/8 can serve as the probability density function of a continuous random variable which can take on any value in the interval from 0 to 4.