Question Bank  Probability  Swetabh Kumar

Number of Questions  10
Topic  Probability
Answer key available  Yes (Last post)
Source 

Q1) An experiment succeeds twice as often as it fails. What is the probability that in the next 5 trials there will be three successes (in any order)?

Q2) Two 1*1 squares are chosen at random on a chessboard. What is the probability that they have a side in common?

Q3) You go to see the doctor about an ingrowing toenail. The doctor selects you at random to have a blood test for swine flu, which for the purposes of this exercise we will say is currently suspected to affect 1 in 10,000 people in Australia. The test is 99% accurate, in the sense that the probability of a false positive is 1%. The probability of a false negative is zero. You test positive. What is the new probability that you have swine flu?

Q4) Ramesh plans to order a birthday gift for his friend from an online retailer. However, the birthday coincides with the festival season during which there is a huge demand for buying online goods and hence deliveries are often delayed. He estimates that the probability of receiving the gift, in time, from the retailers A, B, C and D would be 0.6, 0.8, 0.9 and 0.5 respectively. Playing safe, he orders from all four retailers simultaneously. What would be the probability that his friend would receive the gift in time?

Q5) Bag I contains 4 white and 6 black balls while another Bag II contains 4 white and 3 black balls. One ball is drawn at random from one of the bags and it is found to be black. Find the probability that it was drawn from Bag I.

Q6) In a certain day care class, 30% of the children have grey eyes, 50% of them have blue and the other 20%'s eyes are in other colors. One day they play a game together. In the first run, 65% of the grey eye ones, 82% of the blue eyed ones and 50% of the children with other eye color were selected. Now, if a child is selected randomly from the class, and we know that he/she was not in the first game, what is the probability that the child has blue eyes?

Q7) While watching a game of Champions League football in a cafe, you observe someone who is clearly supporting Manchester United in the game. What is the probability that they were actually born within 25 miles of Manchester ? Assume that:
 the probability that a randomly selected person in a typical local bar environment is born within 25 miles of Manchester is 1/20 , and;
 the chance that a person born within 25 miles of Manchester actually supports United is 7/10 ;
 the probability that a person not born within 25 miles of Manchester supports United with probability 1/10 .

Q8) Five people entered the lift cabin on the ground floor of an 8floor building (this includes the ground floor). Suppose each of them, independently and with equal probability, can leave the cabin at any of the other seven floors. Find out the probability of all five people leaving at different floors.

Q9) A fair die is tossed twice in succession. Let A denote the first score and B the second score. Consider the quadratic equation x^2 + Ax + B = 0. Find the probability that the equation has 2 distinct real roots

Q10) A multiplechoice test consists of five choices per question. You think you know the answer for 75% of the questions and for the other 25% you guess at random. When you think you know the answer, you are right only 80% of the time. Find the probability of getting an arbitrary question right.

Answers:
 OA: 80/243
5C3 * (2/3)^3 * (1/3)^2  OA: 1/18
Number of rectangles of size a * b in M * N grid is (Ma+1)(Nb+1)
here M=8, N=8. just find rectangles of 2 * 1 and 1 * 2.
so (82+1)(81+1) + (81+1)(82+1) = 112.
so 112/64C2 = 1/18  OA: 0.01 (Basic Bayes formula)
 1  (0.4 * 0.2 * 0.1 * 0.5) = 0.996 (XAT 2015)
 0.6/ (6/10 + 3/7) = 7/12 (Bayes theorem)
 OA: 18/59
 7/(7 + 19) = 7/26 (Bayes theorem)
 OA: 7P5/7^5 = 360/2401
 A^2 > 4B
A=1, B=no value
A=2 B=none
A=3 B=1,2
A=4, B=1,2,3
A=5, B=1,2,3,4,5,6
A=6 B=1,2,3,4,5,6. Total : 17 cases. so 17/36  OA: (3/4)(4/5)+(1/4)(1/5)=3/5+1/20 = 13/20 = 0.65
 OA: 80/243