Big Board Plinko Probabilities based on Pascal's Triangle Chip Starting in Slot  A  B  C  D    Summary

Chip starting in Slot A:

 A B C D E F G 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 A' B' C' D' E' F' G' \$ 1 0 0 \$ 5 0 0 \$ 1 0 0 0 \$ 0 \$ 1 0 0 0 0 \$ 0 \$ 1 0 0 0

Note:  Columns H' and I' are omitted.  The probability that a chip starting in A will land in those slots is zero.

Using the reflection technique, we obtain the numerators of the fractions in the chart below.  The denominator is 212 = 4096.

Conditional Probabilities:

 P(A'|A) P(B'|A) P(C'|A) P(D'|A) P(E'|A) P(F'|A) P(G'|A) P(H'|A) P(I'|A) 0 0 0.226 0.387 0.242 0.107 0.032 0.006 0.000 0 0

Chip starting in Slot B:

 A B C D E F G I 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 A' B' C' D' E' F' G' H' \$ 1 0 0 \$ 5 0 0 \$ 1 0 0 0 \$ 0 \$ 1 0 0 0 0 \$ 0 \$ 1 0 0 0 \$ 5 0 0
NOTE:  Column H' is omitted.  The probability that a chip starting in slot B will land in H' is zero.

Using the reflection technique, we obtain the numerators of the fractions in the chart below.  The denominator is 212 = 4096.

Conditional Probabilities:

 P(A'|B) P(B'|B) P(C'|B) P(D'|B) P(E'|B) P(F'|B) P(G'|B) P(H'|B) P(I'|B) 0 0.193 0.346 0.247 0.137 0.057 0.016 0.003 0.000 0

Chip starting in Slot C:

 A B C D E F G H I 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 A' B' C' D' E' F' G' H' I' \$ 1 0 0 \$ 5 0 0 \$ 1 0 0 0 \$ 0 \$ 1 0 0 0 0 \$ 0 \$ 1 0 0 0 \$ 5 0 0 \$ 1 0 0

Using the reflection technique, we obtain the numerators of the fractions in the chart below.  The denominator is 212 = 4096.

Conditional Probabilities:

 P(A'|C) P(B'|C) P(C'|B) P(D'|C) P(E'|C) P(F'|C) P(G'|C) P(H'|C) P(I'|C) 0.121 0.247 0.241 0.196 0.121 0.054 0.016 0.003 0.000

Chip starting in Slot D:

 A B C D E F G H I 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 A' B' C' D' E' F' G' H' I' \$ 1 0 0 \$ 5 0 0 \$ 1 0 0 0 \$ 0 \$ 1 0 0 0 0 \$ 0 \$ 1 0 0 0 \$ 5 0 0 \$ 1 0 0

Using the reflection technique, we obtain the numerators of the fractions in the chart below.  The denominator is 212 = 4096.

Conditional Probabilities:

 P(A'|D) P(B'|D) P(C'|D) P(D'|D) P(E'|D) P(F'|D) P(G'|D) P(H'|D) P(I'|D) 0.054 0.137 0.196 0.226 0.193 0.121 0.054 0.016 0.003

The conditional probabilities for a chip dropped in slots F, G, H, and I are obtained by symmetry.

Summary of Conditional Probabilities:

 Column End Begin A' \$100 B' \$500 C' \$1000 D' \$0 E' \$10000 F' \$0 G' \$1000 H' \$500 I' \$100 A 0.226 0.387 0.242 0.107 0.032 0.006 0.000 0 0 B 0.193 0.346 0.247 0.137 0.057 0.016 0.003 0.000 0 C 0.121 0.247 0.241 0.196 0.121 0.054 0.016 0.003 0.000 D 0.054 0.137 0.196 0.226 0.193 0.121 0.054 0.016 0.003 E 0.016 0.057 0.121 0.193 0.226 0.193 0.121 0.057 0.016 F 0.003 0.016 0.054 0.121 0.193 0.225 0.196 0.137 0.054 G 0.000 0.003 0.016 0.054 0.121 0.196 0.241 0.247 0.121 H 0 0.000 0.003 0.016 0.057 0.137 0.247 0.346 0.193 I 0 0 0.000 0.006 0.032 0.107 0.242 0.387 0.226