The Pascal’s Triangle
Math/Grade 7
Pascal’s Triangle and Events with
2 Possible Outcomes
Line 0
Line 1
Pascal’s Triangle is useful in solving many types of problems. One example is solving for probabilities in which
an event has only two outcomes; for example, when solving problems with coins, true/false questions,
having boys and girls, etc… It is particularly helpful when these events are occurring multiple times.
1. What is the next line in the Triangle? How did you find the numbers?
2. Make a tree diagram for the possible outcomes if two coins are flipped.
a. How many possible outcomes are there?
b. P(2 head) =
c. P(1 head) =
d. P(0 heads) =
e. Do you see a connection between the numbers that you have calculated and Pascal’s
Triangle above?
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
The Pascal’s Triangle
Math/Grade 7
3. Make a tree diagram for the possible outcomes if three coins are flipped.
a. How many possible outcomes are there?
b. P(3 heads) =
c. P(2 heads) =
d. P(1 head) =
e. P(0 heads) =
f. Do you see a connection between the numbers that you have calculated and Pascal’s
Triangle above?
g. P(at least 1 heads) =
3. Using what you have learned form #1 and 2, solve the following based on the flipping of four coins.
a. Which line of Pascal’s Triangle would you use? List the numbers from that line.
b. How many possible outcomes will there be?
c. P(4 heads) =
d. P(3 heads) =
e. P(2 heads) =
f. P(1 head) =
g. P(0 heads) =
h. P(at least 2 heads) =
i.
P(no more than 1 head) =
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
The Pascal’s Triangle
Math/Grade 7
4. A family has 6 children.
a. P(all 6 are girls) =
b. P(5 girls) =
c. P(4 girls) =
d. P(3 girls) =
e. P(4 or more girls) =
f. P(all 6 are boys) =
5. Look at the sums of each line.
a. What pattern do you see?
b. What would be the sum of the tenth line?
c. How would you find the sum of any line in triangle?
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
The Pascal’s Triangle
Math/Grade 7
Pascal’s Triangle and Events with 2 Possible Outcomes - Key
Line 0
Line 1
Pascal’s Triangle is useful in solving many types of problems. One example is solving for probabilities
in which an event has only two outcomes. For example, when solving problems with coins, true/false
questions, having boys and girls, etc… It is particularly helpful when these events are occurring
multiple times.
1. What is the next line in the Triangle? How did you find the numbers?
The next line is 1, 7, 21, 35, 35, 21, 7, 1
The first and last numbers are 1. The numbers in between are the sum of the number just to the left
and right in the line above.
2. Make a tree diagram for the possible outcomes if two coins are flipped.
H
H
T
H
T
T
a. How many possible outcomes are there? 4
b. P(2 head) = 1/4
c. P(1 head) = 2/4
d. P(0 heads) = 1/4
e. Do you see a connection between the numbers that you have calculated and Pascal’s
Triangle above? The numerators 1, 2, 1 are the numbers in line 2 of the triangle. The sum of
these numbers is 4 which is the denominator.
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
The Pascal’s Triangle
Math/Grade 7
3. Make a tree diagram for the possible outcomes if three coins are flipped.
H
H
H
T
H
T
H
T
T
H
T
T
H
T
a. How many possible outcomes are there?
b. P(3 heads) = 1/8
c. P(2 heads) = 3/8
d. P(1 head) = 3/8
e. P(0 heads) = 1/8
f. Do you see a connection between the numbers that you have calculated and Pascal’s
Triangle above? The numerators 1, 3, 3, 1 are the numbers in line 3. The sum of these
numbers is 8.
g. P(at least 1 heads) = 7/8
3. Using what you have learned form #1 and 2, solve the following based on the flipping of four coins.
a. Which line of Pascal’s Triangle will you be using for four coins? Line 4: 1, 4, 6, 4, 1
b. How many possible outcomes will there be? 16
c. P(4 heads) = 1/16
d. P(3 heads) = 4/16
e. P(2 heads) = 6/16
f. P(1 head) = 4/16
g. P(0 heads) = 1/16
h. P(at least 2 heads) = 11/16
i.
P(no more than 1 head) = 5/16
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
The Pascal’s Triangle
Math/Grade 7
4. A family has 6 children. Use Line 6: 1, 6, 15, 20, 15, 6, 5, 1
a. P(all 6 are girls) = 1/64
b. P(5 girls) = 6/64
c. P(4 girls) = 15/64
d. P(3 girls) = 20/64
e. P(4 or more girls) = 22/64
f. P(all 6 are boys) = 1/64
5. Look at the sums of each line.
a. What pattern do you see?
The sum in each line is a power of 2.
b. What would be the sum of the tenth line?
210 = 1024
c. How would you find the sum of any line in triangle?
The sum of line n of the triangle would be 2n.
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project