Question

A salesman receives a commission for each sale he makes together with a fixed daily income. The number of sales he makes in a day along with their probabilities are given in the table below : | X | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---| | P(X) | 0.42 | 3k | 0.3 | 0.05 | 2k | 0.03 | His daily income Y (in ₹) is given by : Y = 800X + 50

Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of the random variable.

A and B throw a die alternately till one of them gets a '6' and wins the game. Find their respective probabilities of wining, if A starts the game first.

An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X. | X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---| | P(X) | p | 2p | 2p | p | 2p | p² | 2p² | 7p² + p |