Random variables : Exercises

Introduction

A simple example problem involving the construction of random variables together with a simple varient on the problem for you to try.

Example problems

Click on the problems to reveal the solution

Problem 1

An urn contains 3 balls a red one, a green one and a blue one. Consider an experiment in which balls are drawn at random and then replaced. Define the sample space for such an experiment if this is done twice. What is the probability for each element in the sample space?

The sample space is:

red red red green red blue
green red green green green blue
blue red blue green blue blue

The classical interpretation of probability means all combinations are equally likely. Hence: $$ P = \frac{1}{9} \nonumber $$

Problem 2

Consider the problem described in the example but now suppose that the balls are not replaced. Enumerate the possible outcomes and calculate the probabilities.

$$ P = \frac{1}{6} $$

Problem 3

What is the probabilities that two randomly selected people have their birthday on the 4th October (assume no leap years)?

$$ P = \left( \frac{1}{365} \right)^2 $$

What is the probability that two randomly selected people have the same birthday (assume no leap years again)?

$$ P = \left( \frac{1}{365} \right) $$

Problem 4

In the game of bridge 4 players share out the 52 cards in a deck. In each round three players play while the last player (the dummy) lays his cards out on the table for this partner (the declarer) to play. If the declarer can see that he and the dummy have 8 of the heart cards between them he knows that the two hands that he cannot see must hold five heart cards between them. What are the probabilities that the cards are split 0:5, 1:4 and 2:3 between the other two players?

  • Probability of 0:5 split : 0.06
  • Probability of 1:4 split : 0.32
  • Probability of 2:3 split : 0.62

Contact Details

School of Mathematics and Physics,
Queen's University Belfast,
Belfast,
BT7 1NN

Email: g.tribello@qub.ac.uk
Website: mywebsite