2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
IT’S GAME TIME: LINK PROBABILITY TO PASCAL THROUGH PLINKO
Plinko is a famous stage game on the television game show, The Price is Right. It
is a game of chance that involves probabilities associated with the contestant
choice of where to drop a chip down a board to land in a slot that will earn the
contestant a given amount of money (Biesterfeld 2001).
For our lesson, we used a smaller version of the plinko board which we adapted
from the Demos for Postive Impact Project website
www.mathdemos.org/mathdemos/plinko/ . This board has three starting slots (A,
B, and C) and three ending slots (A’, B’ and C’). Students will have a chance of
winning $100 or $200 or $5000.
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Nareshn2@muohio.edu
2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
A SMALLER VERSION OF THE GAME OF PLINKO
FOCUS QUESTIONS
1. What are your chances of winning $5000 on this board?
2. If you were given only 1 chip to play on the small Plinko board, which slot
would you drop the chip in? Explain why you selected that slot.
PLINKO SIMULATION PART 1: EXPERIMENTAL PROBABILITY
SIMULATION USING THE ACTUAL BOARD
Number of Trials:5
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2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
CALCULATOR SIMULATION
Focus question 1: What is the experimental probability of winning $5000?
Experimental
Probability
Ending slot
Starting Slot
$100
$5000
$200
$100
$5000
$200
A (30 trials)
8
18
4
8/30
(0.27)
18/30
(0.6)
4/30
(0.13)
B (30 trials)
8
16
6
8/30
(0.27)
16/30
(0.53)
6/30
(0.2)
C (30 trials)
4
10
16
4/30
(0.13)
10/30
(0.33)
16/30
(0.53)
Experimental probability of winning $5000 with a random starting slot
= 1/3 (0.6 + 0.53 + 0.33) = 0.49
Theoretical Probability Table
Ending slot
Theoretical Probability
Starting Slot
$100
$5000
$200
$100
$5000
$200
A
2
3
1
0.125
0.5
0.375
B
3
6
3
0.25
0.5
0.25
C
1
3
2
0.375
0.5
0.125
For an explanation of the theoretical probability values, please read on….
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2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
NUMBER OF PATHS FROM B TO A’
NUMBER OF PATHS FROM B TO B’
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2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
NUMBER OF PATHS FROM B TO C’
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Nareshn2@muohio.edu
2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
THEORETICAL PROBABILITIES FOR SMALL PLINKO BOARD USING TREE DIAGRAMS
Source: http://www.mathdemos.org/mathdemos/plinko/
Conditional Probabilities with B as the starting slot
Here is the board with only three starting and ending slots.
If a chip is dropped from slot B, it can fall left or right. Thus, the probability it will fall to the left
is 1/2 and the probability that it will fall to the right is 1/2. We can illustrate this using a tree
diagram that we construct as in Figure 3. We label each branch of the tree with the probability
that the chip will take a path along that branch. This is stage 1 of the tree.
Figure 3. Stage 1: Tree diagram illustrating the
possible paths and probabilities from slot B to 2nd
row of slots.
Continuing in the same manner, we see that there are two choices for each of the two
possibilities in Step 1, each occurring with probability 1/2. This is illustrated in Figure 4.
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2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
Figure 4. Stage 2: Tree diagram illustrating the
possible paths and probabilities from slot B to 3nd
row of slots.
At this point in the chip's path, there is a possibility that the chip will hit the wall of the game
board if it takes one of the paths shown in red in Figure 5. If the chip encounters the wall of the
game board, it will, with probability 1, fall to the right (if it hits the left side) or fall to the left (if
it hits the right side).
Figure 5. Paths which bring the chip in contact with the wall are
shown in red.
Continuing in the manner illustrated in these examples, we can construct the entire tree
diagram for the small Plinko board. The probability that a chip falling from slot B will land in a
particular slot along a particular path is computed by multiplying the probabilities along the
path. This is the multiplicative property of probability tree diagrams.
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2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
Figure 6. Tree diagram for small Plinko board.
The theoretical probability (this is a conditional probability) of a chip landing in one of the
slots, given that it starts in B is calculated by adding the probabilities for each path from B to
the slot. This is the additive property of probability tree diagrams.
We denote the conditional probability that the chip will land in slot A' given that it starts in B by
P(A' | B) and it is computed by adding the probabilities associated with each path from B to A':
P(A' | B) = 1/8 + 1/16 + 1/16 = 1/4.
Similarly, the probability of the chip falling into slot B’ given that it started in B is
P(B' | B) = 8/16 = 1/2, and
Similarly, the probability of the chip falling into slot C’ given that it started in B is
P(C' | B) = 4/16 = 1/4.
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2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
Conditional probabilities with A as the starting slot
First compute the probabilities for a chip dropped in slot A using a tree diagram.
So
The probability of the chip falling into slot A’ given that it started in A is
P(A' | A) = 1/4 + 1/8 = 3/8
The probability of the chip falling into slot B’ given that it started in A is
P(B' | A) = 1/4 + 1/8 + 1/8 = 1/2
The probability of the chip falling into slot C’ given that it started in A is
P(C' | A) = 1/8
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Nareshn2@muohio.edu
2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
Conditional probabilities with C as the starting slot
Since the plinko board is symmetrical, we can use the above findings to determine the
following probabilities.
The probability of the chip falling into slot A’ given that it started in C is
P(A' | C) = 1/8
The probability of the chip falling into slot B’ given that it started in C is
P(B' | C) = 1/4 + 1/8 + 1/8 = 1/2
The probability of the chip falling into slot C’ given that it started in C is
P(C' | C) = 1/4 + 1/8 = 3/8
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2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
FOCUS QUESTION 2
What is the probability of winning $5000 on this board?
Random Starting Slot: The discussion above dealt with conditional probabilities, for example,
the probability of the chip landing in slot B' given that it started in B. Suppose the player starts
from slot that is randomly selected. How can we compute the probability of the chip falling in a
particular ending slot?
That question can be answered with the help of the multiplicative and additive properties of
tree diagrams. In this case the tree is somewhat different because when the play begins, we
have three choices that can be made (starting slots A, B, or C), each with probability 1/3. So the
tree diagram begins with three branches, one to each of A, B, or C. The rest of the tree is filled
out with the tree diagrams we discussed earlier.
The probabilities can now be calculated using the additive property
P(winning $ 100) = P (A') = 1/3 (P(A' | A) + P(A' | B) + P(A' | C)) = 1/3 (3/8 + 1/4 + 1/8) = 1/4
P(winning $ 5000) = P(B') = 1/3 (P(B' | A) + P(B' | B) + P(B' | C)) = 1/3 (1/2 + 1/2 + 1/2) = 1/2
P(winning $ 200) = P(C') = 1/3 (P(C' | A) + P(C' | B) + P(C' | C)) = 1/3 (1/8 + 1/4 + 3/8) = 1/4.
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Nareshn2@muohio.edu
2012 NCTM conference
It’s Game Time: Link Probability to Pascal through Plinko
REFERENCES
Biesterfeld, A. (2001). The Price (or Probability) Is Right. Journal for Statistics
Education, 9(3).
Blits, S. (2007). Come On Down!: Behind the Big Doors at “The Price is Right”.
New York, NY: HarperCollins.
Butterworth, W. & Coe, P. (2002). The Prizes Rite. Math Horizons, 9(3), 2530.
Hastings, K. J. (1997) Probability and Statistics. Reading, MA: AddisonWesley Longman, Inc, 93-95.
Haws, L. (1995). Plinko, Probability and Pascal. Mathematics Teacher, 88(4),
282-285.
Lanier, S. & Barrs S. (2003). Let’s Play Plinko: A Lesson in Simulations and
Experimental Probabilities. Mathematics Teacher, 96(9), 626-633.
Lemon, P. (1997). Pascal’s Triangle- Patterns, Paths, and Plinko.
Mathematics Teacher, 90(4), 270-273.
The Price is Right. (2011). [Photo of “The new Plinko!”]. Retrieved from
http://www.priceisright.com/show/games/plinko.
Mathdemos.org website:
http://www.mathdemos.org/mathdemos/plinko/
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