In a standard carnival game, you toss a coin on a table top marked with a grid of squares. You win if the coin lands without touching or overlapping any lines. In other words, the coin needs to land entirely inside one of the squares.
If the squares measure 1 1/2 inches across and the coin has a diameter of 1 inch, what is the probability you will win? Assume that the coin always lands on the grid
If you "assume that the coin always lands on the grid," the probability of winning is zero.
Assume that the "grid" means "grid of squares" is the collection of all the squares, then we consider one square:
The "safe" area for the centre of the coin is a square of 1/2"x1/2" out of a total of 1.5"x1.5".
Probability is therefore
P(win)
=.5²/1.5²
=0.25/2.25
=1/9
To find the probability of winning in this carnival game, we can calculate the ratio of favorable outcomes to total outcomes.
In this case, a favorable outcome would be when the coin lands entirely inside one of the squares on the grid without touching or overlapping any lines.
To calculate the total number of outcomes, we need to determine how many squares the coin can potentially land in. We can do this by dividing the total area of the entire grid by the area of each square.
Given that the squares measure 1 1/2 inches across, we need to convert this measurement to square inches. Since the squares are square-shaped, their area is given by the formula: (side length) squared.
Converting 1 1/2 inches to a decimal, we have:
1 1/2 inches = 1.5 inches
The area of each square is then:
(1.5 inches) squared = 2.25 square inches
Now, we need to determine the total area of the grid. The carnival game does not specify the size of the grid, so we'll assume it's a standard square grid.
Let's say the grid has n by n squares. The total area of the grid can then be calculated by multiplying the number of rows by the number of columns:
Total area = (n squares) * (n squares) * (2.25 square inches/square)
Next, we need to determine how many squares the coin can potentially land in without touching or overlapping any lines. Since the coin has a diameter of 1 inch, we need to calculate its area:
Coin area = (1 inch / 2) squared * π (pi)
= 0.25 square inches * 3.14
= 0.785 square inches
Finally, the probability of winning can be calculated as the ratio of favorable outcomes (number of squares the coin can land in) to total outcomes (total number of squares on the grid):
Probability of winning = (Favorable outcomes) / (Total outcomes)
= (Coin area) / (Total area)
Substituting the values we've calculated, we can find the probability of winning.