Assume that there are one red ball and three black balls, and the balls are lined up randomly.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Problem 1

Assume that there are one red ball and three black balls, and the balls are lined up randomly. Define the value of random variable for k-th ball (k=1, 2, 3, 4) as follows:

X = 0 if the color of the k-th ball is black

1 if the color of the k-th ball is red.

(1) Show the joint probability function of X1, X2, X3, X4 is the same. Example : If , we get (X1, X2, X3, X4) = (0, 1, 0, 0)

(2) Explain that the probability of winning one Amidakuji is irrelevant to the order in which you choose. (Amidakuji is also called a ladder lottery.)

Problem 2

In a sequence of characters “a” and “b” randomly arranged, let X be the number of characters until the sequence “ab” first appears, and let Y be the number of characters until the sequence “aa” first appears. What is the magnitude relationship between the expected values of the numbers of X and Y?

Problem 3

A product is produced from two factories A and B. At Factory A, the percentage of defective products produced is 5 %. At Factory B, the percentage of defective products produced is 2 %. The shipment volume of Factory A is twice the shipment volume of Factory B. At this time, calculate the probability that the defective product found in the market is a product produced in Factory A.

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