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MATH / A game of Plinko is played by dropping a ball at the top-middle of a set of…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem
A game of Plinko is played by dropping a ball at the top-middle of a set of pegs. Each time the ball hits a peg, it has a probability of deflecting either to the left or to the right. What is the probability that the ball lands in the middle slot as shown in the diagram (the red line)? [asy]
size(3inch, 2.25inch);
for (int i = 0; i < 8; i += 2) {
draw(circle((0, i + 1), 0.1));
draw(circle((2, i + 1), 0.1));
draw(circle((4, i + 1), 0.1));
draw(circle((6, i + 1), 0.1));
draw(circle((8, i + 1), 0.1));
draw(circle((10, i + 1), 0.1));
draw(circle((12, i + 1), 0.1));
draw(circle((14, i + 1), 0.1));
draw(circle((16, i + 1), 0.1));
draw(circle((1, i), 0.1));
draw(circle((3, i), 0.1));
draw(circle((5, i), 0.1));
draw(circle((7, i), 0.1));
draw(circle((9, i), 0.1));
draw(circle((11, i), 0.1));
draw(circle((13, i), 0.1));
draw(circle((15, i), 0.1));
}
pen target = red + linewidth(2);
draw((-0.8, -1)--(0.8, -1));
draw((1.2, -1)--(2.8, -1));
draw((3.2, -1)--(4.8, -1));
draw((5.2, -1)--(6.8, -1));
draw((7.2, -1)--(8.8, -1), target);
draw((9.2, -1)--(10.8, -1));
draw((11.2, -1)--(12.8, -1));
draw((13.2, -1)--(14.8, -1));
draw((15.2, -1)--(16.8, -1));
pen tube = black + linewidth(2);
draw((7.2, 9)--(7.2, 7.8)--(8.8, 7.8)--(8.8, 9), tube);
filldraw(Circle((8, 8.5), 0.6), blue);
[/asy]
Plain-text mathematical notation (without MathML)
A game of Plinko is played by dropping a ball at the top-middle of a set of pegs. Each time the ball hits a peg, it has a (1)/(2) probability of deflecting either to the left or to the right. What is the probability that the ball lands in the middle slot as shown in the diagram (the red line)? [asy]
size(3inch, 2.25inch);
for (int i = 0; i < 8; i += 2) {
draw(circle((0, i + 1), 0.1));
draw(circle((2, i + 1), 0.1));
draw(circle((4, i + 1), 0.1));
draw(circle((6, i + 1), 0.1));
draw(circle((8, i + 1), 0.1));
draw(circle((10, i + 1), 0.1));
draw(circle((12, i + 1), 0.1));
draw(circle((14, i + 1), 0.1));
draw(circle((16, i + 1), 0.1));
draw(circle((1, i), 0.1));
draw(circle((3, i), 0.1));
draw(circle((5, i), 0.1));
draw(circle((7, i), 0.1));
draw(circle((9, i), 0.1));
draw(circle((11, i), 0.1));
draw(circle((13, i), 0.1));
draw(circle((15, i), 0.1));
}
pen target = red + linewidth(2);
draw((-0.8, -1)--(0.8, -1));
draw((1.2, -1)--(2.8, -1));
draw((3.2, -1)--(4.8, -1));
draw((5.2, -1)--(6.8, -1));
draw((7.2, -1)--(8.8, -1), target);
draw((9.2, -1)--(10.8, -1));
draw((11.2, -1)--(12.8, -1));
draw((13.2, -1)--(14.8, -1));
draw((15.2, -1)--(16.8, -1));
pen tube = black + linewidth(2);
draw((7.2, 9)--(7.2, 7.8)--(8.8, 7.8)--(8.8, 9), tube);
filldraw(Circle((8, 8.5), 0.6), blue);
[/asy]Original LaTeX notation
A game of Plinko is played by dropping a ball at the top-middle of a set of pegs. Each time the ball hits a peg, it has a $\frac{1}{2}$ probability of deflecting either to the left or to the right. What is the probability that the ball lands in the middle slot as shown in the diagram (the red line)? [asy]
size(3inch, 2.25inch);
for (int i = 0; i < 8; i += 2) {
draw(circle((0, i + 1), 0.1));
draw(circle((2, i + 1), 0.1));
draw(circle((4, i + 1), 0.1));
draw(circle((6, i + 1), 0.1));
draw(circle((8, i + 1), 0.1));
draw(circle((10, i + 1), 0.1));
draw(circle((12, i + 1), 0.1));
draw(circle((14, i + 1), 0.1));
draw(circle((16, i + 1), 0.1));
draw(circle((1, i), 0.1));
draw(circle((3, i), 0.1));
draw(circle((5, i), 0.1));
draw(circle((7, i), 0.1));
draw(circle((9, i), 0.1));
draw(circle((11, i), 0.1));
draw(circle((13, i), 0.1));
draw(circle((15, i), 0.1));
}
pen target = red + linewidth(2);
draw((-0.8, -1)--(0.8, -1));
draw((1.2, -1)--(2.8, -1));
draw((3.2, -1)--(4.8, -1));
draw((5.2, -1)--(6.8, -1));
draw((7.2, -1)--(8.8, -1), target);
draw((9.2, -1)--(10.8, -1));
draw((11.2, -1)--(12.8, -1));
draw((13.2, -1)--(14.8, -1));
draw((15.2, -1)--(16.8, -1));
pen tube = black + linewidth(2);
draw((7.2, 9)--(7.2, 7.8)--(8.8, 7.8)--(8.8, 9), tube);
filldraw(Circle((8, 8.5), 0.6), blue);
[/asy]level
Level 5
type
Counting & Probability
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initial import