Harry Marshall
Feb 7 Journal
Group B
Assignment: Lotto 6/49
Given 49 different numbers: 1 2 3 … 49
players choose 6 numbers on their ticket
Random selection of 6 (order not important) as winner
1st prize: all 6 correct 2nd prize: 5 out of 6 correct
Calculate the probability of winning 1st prize.
How many different tickets? 49C6 = 13,983,816
How many ways to win?
1
probability of 1st prize? 1 out of 13,983,816
2nd prize: 5 out of 6 correct
How many different tickets? 49C6 = 13,983,816
How many ways to win?
6C5 X 43C1 =258
probability of 2nd prize?
258 out of 13,983,816 or 1.845 X 10^5
Feb 7 Assignment: more choices
3rd prize: 4 out of 6 correct
How many different tickets? 49C6 = 13,983,816
How many ways to win?
6C4 X 43C2 = 13,545
probability of 2nd prize?
.1% or 1 out of 1,000)
13,545 out of 13,983,816 or .00096862 or (around
Assignment: Basketball:
A basketball team has 10 players. During each practice the coach randomly divides the
players into 2 5-player teams. What is the probability that identical teams are chosen in
two successive practices?
Number of possible teams: 10C5
=
252 leaving the other 5 as the other team.
So the chances of identical teams in two successive practices is 1 out of 252.
1
Harry Marshall
Feb 7 Journal
Group B
If the team has 12 players and the coach randomly chooses 2 5-player teams and 2
referees at each practice, what is the probability that identical teams are chosen in two
successive practices? find 3 different methods.
Method 1: Choosing both teams with those not chosen becoming referees:
12C5 X 7C5 = 16,632; therefore the probability is 1 out of 16,632.
Method 2: Choosing the two refs, then the first team, with the 5 players remaining
becoming the second team:
12C2 X 10C5 = 16,632; therefore the probability is 1 out of 16,632.
Method 3: Choosing one team, then the two refs, with the 5 players remaining becoming
the second team:
12C5 X 7C2 = 16,632, therefore the probability is 1 out of 16,632.
Design 2 similar problems that would be a challenge, but not impossible, for most of your
students.
1. Mr. Marshall divides his class of twenty students into two groups of ten that are
each responsible for a PowerPoint presentation. How many possibilities are there
for the two groups?
20C10 =
184,786
If Mr. Marshall divides these students randomly into groups once at the beginning
of the year and once at mid-terms (and the class has the same students), what is
the probability the groups will be exactly the same at mid-terms as they are at the
start of the year?
1 out of 184,786
2. What if Mr. Marshall divides the class into four groups of five students each?
How many possibilities are there for the four groups?
20C5
X 15C5 X 10C5 =11,732,745,024
If Mr. Marshall divides these students randomly into groups once at the beginning
of the year and once at mid-terms (and the class has the same students), what is
the probability the groups will be exactly the same at mid-terms as they are at the
start of the year?
1 out of 11,732,745,024
2
Harry Marshall
Feb 7 Journal
Group B
Assignment: more one way Manhattan patterns
What is the distance from (0,0) to (i,j) with i≥0 and j≥0 when:
• i and j are even? I+J
• i is even and j is odd? I+J
• i is odd and j is even? I+J
• i and j are odd? I+J+2
3
Harry Marshall
Feb 7 Journal
Group B
What are the reverse distances back to (0,0) for each of the 4 cases?
• i and j are even? I+J+4
• i is even and j is odd? I+J+2
• i is odd and j is even? I+J+2
• i and j are odd? I+J+2
Notice that on the worksheet below, the distance back from any point (i,j) in Quadrant 1
to the origin is the exact same as the distance to the point (-i,-j) in the 4th quadrant from
the origin. See the image below, the fouth quadrant values for distances to (–i,-j) are the
same as the first quadrant values for the return paths from (i,j).
4
Harry Marshall
Feb 7 Journal
Group B
What are the number of paths for each of the 4 cases?
The answer is the same four all 4 cases. In quadrant 1, Paths (i,j) = Paths i-2 + j-2.
5
Harry Marshall
Feb 7 Journal
Group B
Pascal using Combinations Assignment
Construct 2 versions of Pascal’s Δ in Excel using the same =COMBIN( , ) function
in all cells in each Δ.
With cells = 0 formatted to white text inside of borders.
6
Harry Marshall
Feb 7 Journal
Group B
Excel assignments with Manhattan geometry
Complete and document all of the Excel worksheets used in class, including:

plotting individual paths

plotting individual paths from binary expansions
7
Harry Marshall
Feb 7 Journal

Group B
random paths
8
Harry Marshall
Feb 7 Journal
Group B
9