Below are some solved simple exercises and
applications:
1) Find the probability of getting a
multiple of 2 at a die roll.
Solution: The number of outcomes that
are favorable to respective event is three (these are: {2}, {4}, {6}).
The number of equally possible outcomes is six, so the probability is
3/6 = 1/2 = 50%.
2) There are three pairs of socks of
different colors in a basket. Two socks are randomly extracted from the
basket. What is the probability of getting two socks of same color?
Solution:
The number of equally possible cases is the number of all 2size
combinations of socks, namely, C(6, 2) = 15. The number of favorable
cases is three, because we have three pairs of socks having the same
color. Thus, the probability is 3/15 = 1/5 = 20%.
3) An urn contains four white balls and six
black balls. Two balls are drawn simultaneously. Find the probability of
the events: a) A drawing two white balls; b) B drawing two black
balls; c) C drawing two balls of the same color.
Solution: The
number of possible cases is C(10, 2). a)
The number of cases that are favorable to event A is C(4, 2);
therefore,.
b) Similarly, .
c) We have .
The events A and B are incompatible, so
.
4) Two dice, one red and one blue, are
rolled. Consider the events: A occurrence of a number less than 4 on
the red die; B occurrence of a number less than 3 on the blue die.
Find P(A or B).
Solution: The cases that are
favorable to A are {1}, {2} and {3}; therefore, P(A)
= 3/6. The cases that are favorable to B are {1} and {2};
therefore, P(B) = 2/6. The cases that are favorable to
A and B correspond to the ordered pairs (1, 1), (1, 2), (2, 1), (2,
2), (3, 1), (3, 2), totalling six, in a probability field where the
number of equally possible cases is 6
x 6 = 36. We
then have P(A and B) = 6/36. The requested
probability is.
5) At a blackjack game, calculate the probability for a player
to get a total of twenty points from the first two cards (provided no
other cards are shown), if a 52card deck is used.
Solution: The
variants totaling twenty points are of the type A + 9 or 10 + 10 (as a
value; that is, any 2size combination of cards from 10, J, Q,
K). We have sixteen variants A + 9 (4 aces and 4 nines) and C(16,
2) = 120 variants 10 + 10 (all 2size combinations of cards from the
sixteen cards with a value of 10). The number of all possible
distribution variants for two cards is C(52, 2)=1326. The probability is
then P = (16 + 120)/1326 = 68/663.
6) We have two urns, the first containing
three white balls and four black balls and the second three white balls
and five black balls. A ball is drawn from a randomly chosen urn. Find
the probability for the drawn ball to be white.
Solution: Denote
the events: A the first urn is the chosen one; B the
second urn is the chosen one; C the drawn ball is white. A
and B form a complete system of events and P(A) =
P(B) = 1/2. We have P(C│A) =
3/7 and P(C│B) = 3/8. According to total
probability formula, we have: P(C) = P(A)P(C│A)
+ P(B)P(C│B) = (1/2)
x (3/7) + (1/2)
x (3/8) = 45/112
= 0.40178.
7) Five cards are drawn at once from a
32card deck, containing cards from 7's up to aces. What is the
probability of the five cards containing at least one queen (Q)?
Solution:
Denoting by A the event to be measured the five extracted
cards contain at least one Q, we then calculate the probability of
the contrary event
the five extracted cards contain no Q.The equally possible
elementary events are the occurrences of 5size combinations of cards from the 32, a total
of C(32, 5). The combinations that are favorable to event
have
the form (xyztv), with x, y, z, t,
v taking any card as value, except the four Qcards. They
total C(32 4, 5) = C(28, 5). We then have:
.
For a broad range of applications in gambling, see
the
gambling page.
Sources 
In
the book UNDERSTANDING AND CALCULATING THE ODDS:
Probability Theory Basics and Calculus Guide for Beginners, with
Applications in Games of Chance and Everyday Life, the teaching
material is structured for developing probability calculus skills based on
algorithmic procedures. This
is the subject of the chapter titled Beginners
Calculus Guide, in which the reader is taught to apply the properties
of probability and to perform calculations in practical applications.
The skills acquired here can be practiced on
the more than 200 solved and unsolved problems and exercises in the book,
whose difficulty level grows gradually. You may find it in the Books
section with a free sample.

