# Probability [ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)]: Questions 8 - 11 of 72

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## Question number: 8

» Probability » Tchebycheffs Inequality

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Appeared in Year: 2014

Essay Question▾

### Describe in Detail

Show that for 40,000 throws of a balanced coin, the probability is at least 0·99 that the proportion of heads will fall between 0·475 and 0·525.

### Explanation

For balanced coin the probability is p = 0.5. For Bernoulli trails where n = 40,000, the mean and standard deviation is

Using Chebyshev’s inequality,

The question says that the probability is at least 0.99 that is

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## Question number: 9

» Probability » Convergence » In Probability

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Appeared in Year: 2011

Essay Question▾

### Describe in Detail

Let X 1, X 2, …, X n be a sequence of i. i. d. r. v. s with E (X i) = 0 and V (X i) = 1. Show that the sequence tends to 1 in probability.

### Explanation

we have known that is the sample variance of the sequence. The mean is

The convergence in probability is

Using Chebychev’s inequality

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## Question number: 10

» Probability » Bayes' Theorem

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Appeared in Year: 2012

Essay Question▾

### Describe in Detail

The ith box contains 2i white balls and 6 - 2i black balls, i = 1 (1) 3. A fair die is cast once. 3 balls are taken at random from box 1, box 2 or box 3 according as the die shows up face 1, any of 2 and 3, or any of 4,5 and 6, respectively. Let X denotes the number of white balls drawn. Find E (X).

### Explanation

E 1 = Box 1 2 white and 4 black when the fair dice value is x 1 =1

E 2 =Box 2 4 white and 2 black when the fair dice value is x 2 = (2,3)

E 3 =Box 3 6 white when the fair dice value is x 3 = (4,5, 6)

X denotes the number of white balls out of randomly select 3 balls

Probability of selecting a ball in

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## Question number: 11

» Probability » Standard Probability Distributions » Negative Binomial

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Appeared in Year: 2012

Essay Question▾

### Describe in Detail

Items from a large lot are examined one by one until r items with a rare manufacturing defect are found. The proportion of items with this type of defect in the lot is known to be p. Let X denote the number of items needed to be examined. Derive the probability distribution of X, and find E (X).

### Explanation

In this question, the sample size is n = x+r given and each trail only two possible outcomes. The probability of defect is same for each trail and trails are independent. The experiment continues until r defectives.

In the given question the number of manufacturing defect are fixed which is r and proportion of item which is defect that is probabilit

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