Section 4.4: “Or problems”
A problem that involves a single experiment is considered an “OR” probability problem if it
requires obtaining a successful outcome for one or another event.
We will usually see the word “OR” in any problem that is an “OR” probability problem.
However, sometimes we will see the words “at least” and the problem will be solved as an “OR”
problem.
We will use this formula to solve “OR” probability problems.
“OR” probability formula”: P (A or B) = P (A) + P(B) – P (A and B )
(we need to subtract so that outcomes aren’t counted twice).
Example: Suppose we roll one die and we are interested in finding the probability of rolling an
even number OR a number greater than 4.
For this situation rolling EITHER a 2, 4, 6 (an even number) OR a 5, 6 (a number > 4) would be
considered a successful outcome.
This is an OR problem.
This is how the problem might be asked using symbols instead of words.
P (even # or > 4) =
To make our example fit the formula, I need to call one of the desired outcomes A and the
other desired outcome B.
A will be the event that an even number is rolled,
B will be the event that a number greater than 4 is rolled.
P(A or B)
= P(A)
+ P(B)
- P(A and B)
P (even # or > 4) = P (even # ) + P ( # > 4 ) – P ( even # and # > 4)
=
3
6
+
2
6
-
1
6
4
2
= 6 𝑜𝑟 3
(If we don’t subtract out the end piece then the # 6 would get counted twice!)
Answer: 2/3
Example: The numbers 1, 2, 3, 4, …., 10 are placed on a piece of paper and thrown into a
bucket. One piece of paper is randomly selected. What is the probability that the piece of
paper has an even number OR a number greater than six on it?
I will call:
A the event of selecting an even number.
B the event a number greater than 6 is selected
P(A OR B) =
P(A)
+ P(B) - P(A and B)
P (even # or # > 6 ) = P (even #) + P ( # > 6 ) – P ( even # and # > 6 )
5
4
2
=
+ 10
- 10
=
10
7
10
The answer is 7/10
Example: From the problem above, what is the probability that the paper has a number less
than five OR a number greater than eight?
A be the event a number less than 5 is selected
B be the even a number greater than 8 is selected
P(A OR B) =
P(A)
+ P(B)
-
P(A and B)
P ( # < 5 or # > 8 ) = P ( # < 5 ) + P ( # > 8 ) – P ( # < 5 and # > 8 )
4/10
+ 2/10
0/10
Answer: 6/10 = 3/5
Note: In this example the probability of a number being less than 5 AND greater than 8 is zero!!
We say that these two events are mutually exclusive.
Definition: Two events are said to be mutually exclusive if it is impossible for both events to
occur simultaneously.
If I am solving an “OR” problem and I know the two events are mutually exclusive I can skip the
subtraction part of the formula as it equals 0.
Homework
#1-4: One card is selected at random from a standard deck of playing cards. Determine the
probability of:
P (A or B). Hint use the formula: P(A or B) = P(A) + P(B) – P(A and B)
1. A = an ace B = a jack
2. A = a red card
3. A = an ace
B = a black card
B = a diamond
4. A = a picture card (J,Q,K)
B = a red card
Homework #5-8 use the formula P(A or B) = P(A) + P(B) – P(A and B) to answer the following.
5) If P(A) = 0.5 and P(B) = 0.6 and P(A and B) = 0.2 find P(A or B)
6) If P(A) = 0.6 and P(B) = 0.8 and P(A and B) = 0.5 find P(A or B)
7) If P(A or B) = 0.8, P(A) = 0.5, P(B) = 0.6 find P(A and B)
8) If P(A or B) = 0.6 and P(B) = 0.3 and P(A and B) = 0.1 find P(A)
Homework: #9 – 12: a single dice is rolled one time. Find the probability of rolling
9) An odd number or a number greater than 2
10) An even number or a number less than 5
11) A number less than 3 or greater than 5
12) A number greater than 5 or less than 4
Homework #13 – 18: One card is selected from a deck of playing cards. Find the probability of
selecting
13) A jack or a diamond
14) A queen or a red card
15) A seven or a six
16) A queen or a two
17) A red card or a ten
18) A black card or a queen
Homework #19 – 22: One marble is selected from a bag that contains 4 red, 5 white, 6 blue and
2 yellow. Find the probability of selecting
19) A red or a white
20) A red or a blue
21) A blue or a white
22) A yellow or a blue
Homework #23 – 26: The numbers 1, 2, 3, 4, …., 14 are placed on a piece of paper and thrown
into a bucket. One piece of paper is randomly selected. What is the probability that:
23) The number is odd or greater then 7
24) The number is even or less than 8
25) The number is greater than or equal to 5 or even
26) The number is less than or equal to 8 or odd
Homework #27-30: The letters of the word Mississippi are placed on pieces of paper and
thrown into a bucket. One piece of paper is randomly selected. What is the probability that:
27) The letter is a p or an m?
28) The letter is a vowel or an m?
29) The letter is a vowel or alphabetically after r?
30) The letter is a vowel or alphabetically before r?
Answers:
1) 8/52 or 2/13 3) 16/52 = 4/13 5) 0.9 7) 0.3 9) 5/6 11) 3/6 = ½ 13) 16/52 = 4/13
15) 8/52 = 2/13 17) 28/52 = 7/13 19) 9/17 21) 11/17 23) 11/14 25) 12/14 = 6/7
27) 3/11 29) 8/11