MATH 109
Basic Probability
We will now formally measure the chance of a random event happening. For example,
what is the probability of winning the Powerball lottery? If two dice are rolled, then
what are the chances of rolling a 7 or an 11?
In order to determine probabilities of random events, we must be able to count the
number of possible outcomes. Assuming first that each outcome is equally likely, then
we can state the following definition:
Let Ω be a set with N elements. If we choose one element at
random, then each element is equally likely to be chosen. In this
case, the probability of any single element being chosen is 1 / N .
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Example 1. Draw a card at random from a shuffled deck. €What is the probability of
drawing the Queen of Hearts?
Answer. Since there are 52 cards, the probability is 1/52.
We also can find the probability of a group of outcomes occurring. We state the
result as follows:
Let A be a subset of Ω having k elements. Choose a single
element at random from the entire set Ω . The probability that this
element is from subset A equals k / N .
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Example 2. Draw a single card from a standard
deck. What is the probability of
drawing a Heart?
Answer. Since there are 13 Hearts, the probability is 13/52 = 1/4.
Interpretation
The fact that there 13 Hearts in a deck means that 13 out of 52, or one-fourth of the deck
is a Heart. So exactly 25% of the cards are Hearts. However when you draw one card, it
either is or it is not a Heart. But the probability of drawing a Heart is still 1/4 = 0.25,
which can be interpreted as follows:
When drawing a card over and over from a shuffled deck, then about 25% of the
time you will draw a Heart.
You cannot say that you will draw a Heart exactly 25% of the time. You can only
give an approximate measure of the percentage of times you will draw a Heart
provided you choose cards at random over and over many times.
Example 3. You are playing the Powerball for which you must match 5 numbers chosen
from 1 – 59 (without repeats and without regard to order), and then match the
powerball chosen from 1 – 35. Your group buys 22 different sets of numbers. What is
the probability that your group will win?
Solution. There are C(59, 5) × 35 = 175,223,510 possible ways to pick the numbers. Since
the drawing is random, we may assume that each individual pick is equally likely to be
1
the winner; thus, the probability of any single pick being the winner is
.
175,223,510
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The probability of the winning pick being one of your group’s 22 picks is
22
1
which simplifies to
. So you have slightly
better than a 1 in 8
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173,223,510
7,964,705
million chance of winning when buying 22 tickets with distinct picks of numbers.
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€ Committee is holding a forum on “religious freedom.” At
Example 4. The Diversity
one meeting, there are 8 Catholics, 12 Protestants, 6 Jews, 6 Muslims, and 4 Hindus. A
Herald reporter picks one of these attendees at random for a quote. What is the
probability that the person is
(a) Muslim?
(c) Christian?
(b) not a Protestant?
(d) neither Christian nor Jewish?
(e) Explain the meaning of the result in (a).
Solution. There are 8 + 12 + 6 + 6 + 4 = 36 persons altogether at this particular meeting.
Assuming the reporter truly picks a person at random (perhaps the first one who
happens to leave the room), then each individual has probability 1/36 of being chosen
by the reporter.
(a) There are 6 Muslims; the probability of a Muslim being asked is 6/36 = 1/6.
(b) There are 24 who are not Protestants, so the probability of not choosing a Protestant
is 24/36 = 2/3.
(c) Since both Catholics and Protestants are considered Christian and there are 20 of
these altogether, the probability is 20/36 = 5/9.
(d) Excluding the 20 Christians and 6 Jews, there are 10 remaining. So the probability is
of choosing neither a Christian nor Jewish person is 10/36 = 5/18.
(e) If a reporter picks one person at random from this group, over and over, many times,
then about 16.67% of the time, the reporter will pick a Muslim.
Dice Probabilities
Roll two dice. How many possible outcomes are there and what are these outcomes?
When considering the sum of the two dice, what is the probability of rolling (i) Doubles
(ii) an even sum? Explain the meaning of each.
When rolling two dice, there are 6 × 6 = 36 possible pairs of outcomes.
outcomes are listed below:
(1, 1)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
These
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
We then add the two dice together to obtain the resulting sum. The possible sums
are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Although each individual paired outcome is equally
likely to occur with probability 1/36, the sums are not equally likely. The sum of 2 can
only occur one way; so its probability is 1/36. But the sum of 7 can occur six ways; so
its probability is 6/36 = 1/6. Below is a chart showing all the various probabilities.
Notice the symmetry as the values increase from 1/36 to 6/36 then back down to 1/36.
Sum
Prob
2
1
36
3
2
36
4
3
36
5
4
36
6
5
36
7
6
36
8
5
36
9
4
36
10
3
36
11
2
36
12
1
36
There are 6 Doubles: (1, 1), (2, 2), . . . , (6, 6). Therefore the probability of rolling
doubles is 6/36 = 1/6 = 0.16666 . So when rolling dice over and over (like when playing
backgammon or monopoly), then about 16 to 17 percent of the time you will roll
doubles.
The even sums are 2, 4, 6, 8, 10, and 12. The overall probability of an even sum is
1 + 3 + 5 + 5 + 3 + 1 18 1
=
= . So when rolling dice over and over, then about 50 percent
36
36 2
of the time you will roll an even sum.
Exercises
(Reduce fractions to simplest form.)
1. A political science class contains 12 Democrats, 10 Republicans, 18 Independents,
and 8 Tea Partiers. If the professor calls on a student at random, then what is the
probability that the student is
(a)
(b)
(c)
(d)
(e)
a Democrat?
not an Independent?
Independent or Tea Partier?
neither a Republican nor a Democrat?
Explain the meaning of the result in Part (a).
2. A sorority has 12 freshmen, 18 sophomores, 20 juniors, and 25 seniors. If a member
is chosen at random, then what is the probability that she is
(a)
(b)
(c)
(d)
(e)
a sophomore?
an upperclassman?
neither a sophomore nor a junior?
not a freshman?
Explain the meaning of the result in Part (b).
3. A deck of cards (including Jokers) has 36 number cards, 12 face cards, 4 aces, and 2
jokers. When picking a card at random, what is the probability of picking
(a)
(b)
(c)
(d)
(e)
a face card
a non-number card
an ace or a joker
neither a face card nor an ace
Explain the meaning of the result in Part (c).
4. You are playing the Lotto South for which you must match 6 numbers chosen from
the numbers 1 – 49 (without repeats and without regard to order). Your group buys 28
different sets of numbers. What is the probability that your group will win?
5. Roll two dice. What is the probability of rolling a 7 or an 11? Explain the meaning.
Answers
1. There are 48 students altogether:
(a) 12/48 = 1/4
(b) 30/48 = 5/8
(c) 26/48 = 13/24
(d) 26/48 = 13/24
(e) When calling on students at random over and over again, then about 25% of the
time the professor will call on a Democrat.
2. There are 75 members altogether:
(a) 18/75 = 6/25
(b) 45/75 = 3/5
(c) 37/75
(d) 63/75 = 21/25
(e) If a member is chosen at random over and over, then about 60% of the time an
upperclassman will be chosen.
3. (a) 12/54 = 2/9
(b) 18/54 = 1/3
(c) 6/54 = 1/9
(d) 38/54 = 19/27
(e) When choosing a card at random over and over, then about 11% of the time, it will
be an Ace or a Joker.
4. There are C(49, 6) = 13,983,816 possible ways to pick the numbers. Since the
drawing is random, we may assume that each individual pick is equally likely to be the
1
winner; thus, the probability of any single pick being the winner is
. So the
13, 983, 816
28
probability of the winning draw being one of your group’s 28 picks is
which
13, 983, 816
1
simplifies to
.
499,422
5. There are 6 × 6 = 36 possibilities for rolling two dice. There are six ways to roll a 7
which are (1, 6), (2, 5), (3, 4), . . . , (6, 1). There are two ways to roll an 11 which are (6, 5)
6+2
8
2
and (5, 6). Therefore the probability of rolling a 7 or an 11 is
=
= = 0.2222 .
36
36 9
So when rolling two dice over and over again, about 22% of the time you will roll a 7
or an 11.