Chapter 27

A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/ loss (in Rs.) of the person is : [April 12, 2019 (II)] (a) \(\frac{1}{2}\) gain (b) \(\frac{1}{4}\) loss (c) \(\frac{1}{2} \operatorname{los} \mathrm{s}\) (d) 2 gain

A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If \(X\) be the number of white balls drawn, then \(\left(\frac{\text { mean of } \mathrm{X}}{\text { standard deviation of } \mathrm{X}}\right)\) is equal to:[Jan. 11, 2019 (II)] (a) 4 (b) \(4 \sqrt{3}\) (c) \(3 \sqrt{2}\) (d) \(\frac{4 \sqrt{3}}{3}\)

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is: [2017] (a) \(\frac{6}{25}\) (b) \(\frac{12}{5}\) (c) 6 (d) 4

An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is: \(\quad\) Online April 10, 2016] (a) \(\frac{496}{729}\) (b) \(\frac{192}{729}\) (c) \(\frac{240}{729}\) (d) \(\frac{256}{729}\)

If the mean and the variance of a binomial variate \(\mathrm{X}\) are 2 and 1 respectively, then the probability that \(X\) takes a value greater than or equal to one is : [Online April 11, 2015] (a) \(\frac{9}{16}\) (b) \(\frac{3}{4}\) (c) \(\frac{1}{16}\) (d) \(\frac{15}{16}\)

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is: [2013] (a) \(\frac{17}{3^{5}}\) (b) \(\frac{13}{3^{5}}\) (c) \(\frac{11}{3^{5}}\) (d) \(\frac{10}{3^{5}}\)

Consider 5 independent Bernoulli's trials each with probability of success \(p\). If the probability of at least one failure is greater than or equal to \(\frac{31}{32}\), then \(p\) lies in the interval (a) \(\left(\frac{3}{4}, \frac{11}{12}\right]\) (b) \(\left[0, \frac{1}{2}\right]\) (c) \(\left(\frac{11}{12}, 1\right]\) (d) \(\left(\frac{1}{2}, \frac{3}{4}\right]\)

In a binomial distribution \(B\left(n, p=\frac{1}{4}\right)\), if the probability of at least one success is greater than or equal to \(\frac{9}{10}\), then \(n\) is greater than: [2009] (a) \(\frac{1}{\log _{10} 4+\log _{10} 3}\) (b) \(\frac{9}{\log _{10} 4-\log _{10} 3}\) (c) \(\frac{4}{\log _{10} 4-\log _{10} 3}\) (d) \(\frac{1}{\log _{10} 4-\log _{10} 3}\)

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is [2007] (a) \(8 / 729\) (b) \(8 / 243\) (c) \(1 / 729\) (d) \(8 / 9\).

At a telephone enquiry system the number of phone calls regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10 minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is (a) \(\frac{6}{5^{\mathrm{e}}}\) (b) \(\frac{5}{6}\) (c) \(\frac{6}{55}\) (d) \(\frac{6}{\mathrm{e}^{5}}\)

A random variable \(X\) has Poisson distribution with mean \(2 .\) Then \(P(X>1.5)\) equals (a) \(\frac{2}{e^{2}}\) (b) 0 (c) \(1-\frac{3}{e^{2}}\) (d) \(\frac{3}{e^{2}}\)

The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is [2004] (a) \(\frac{28}{256}\) (b) \(\frac{219}{256}\) (c) \(\frac{128}{256}\) (d) \(\frac{37}{256}\)

The mean and variance of a random variable \(X\) having binomial distribution are 4 and 2 respectively, then \(P\) \((X=1)\) is [2003] (a) \(\frac{1}{4}\) (b) \(\frac{1}{32}\) (c) \(\frac{1}{16}\) (d) \(\frac{1}{8}\)

A dice is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of success is (a) \(8 / 3\) (b) \(3 / 8\) (c) \(4 / 5\) (d) \(5 / 4\)