Probability
Squares on a chessboard
A5, 2019
You are given an
Solution
Yet another dice roll
A4, 2020
(i) What is the probability that among
(ii) Calculate the probability that out of the last
Solution
(i) The probability that none of the first three rolls have a 4 is
(ii) Let
Let us calculate
Coin toss and a dice roll
B2a, 2021
B2 (a). A mother and two daughters have a fair coin and play a game as follows. First, the mother tosses the coin.
Case I: If the coin lands heads, then both the daughters win.
Case II: If the coin lands tails, then each daughter rolls a fair die independently. The first daughter wins if her die comes up 5 or 6. The second daughter wins if her die comes up 5 or 6.
A game is played.
- What is the probability that the second daughter has lost given that the first daughter has lost?
- What is the probability that the second daughter won given that the first daugher has won?
Solution
- Since the outcome of the toss was tails, the second daugther lost since the die came up 1,2,3 or 4. Hence, required probability is 2/3.
- Let
the event that the second daughter won and be the event that the first daughter won.
A pair of events
A2, 2022
and are mutually exclusive if and only if and are exhaustive. and are independent if and only if and are independent. and cannot be simultaneously independent and exhaustive. and cannot be simultaneously mutually exclusive and exhaustive.
Tinku’s chocolate
A5, 2012
a)
b) Solve the same problem assuming instead that all distributions are equally likely. You are given that the number of such distributions is
The chocolates are considered interchangeable but students are considered different.
Solution
(a) The probability of Tinku not getting a chocolate in one step is
Sol 1. (b) There are
Sol 2. (b) The number of distributions in which Tinku gets no chocolate
Sampling a quadratic
A8, 2013
Consider the quadratic equation
Let
Complete the sentences below.
(a) The equation
(b) The value of
(c) As a function of
(d) As
Solution
(a) The equation
(b) The value of
(c) Increasing. This is because
So
(d) This is the fraction of the area of the unit square
Conditional probability
A11, 2015
There are four distinct balls labelled 1,2,3,4 and four distinct bins labelled
Solution
(i)
Vertex in a polygon
A5, 2014
A regular 100-sided polygon is inscribed in a circle. Suppose three of the 100 vertices are chosen at random, all such combinations being equally likely. Find the probability that the three chosen points form vertices of a right angled triangle.
Solution
Test preparation
B1, 2016
Out of the 14 students taking a test, 5 are well prepared, 6 are adequately prepared and 3 are poorly prepared. There are 10 questions on the test paper. A well prepared student can answer 9 questions correctly, an adequately prepared student can answer 6 questions correctly and a poorly prepared student can answer only 3 questions correctly.
(a) If a randomly chosen student is asked two distinct randomly chosen questions from the test, what is the probability that the student will answer both questions correctly?
(b) Now suppose that a student was chosen at random and asked two randomly chosen questions from the exam, and moreover did answer both questions correctly. Find the probability that the chosen student was well prepared.
Express your answers as irreducible fractions.
Solution
(a) The probability that a randomly chosen student is well prepared is
Let
(b) The probability that a randomly chosen student was well prepared given that he answered both questions correctly is