Probability can be determined based on combinations and permutations. Combinations disregard the order of the numbers, whereas permutations are dependent on order. The code to unlock your phone is an example of a permutation and the different placement of toppings on a hamburger is a combination.
To solve a permutation that allows repetition, the formula is nʳ, where n is the number of options and r is the number of positions or slots to fill. If your phone requires a 4-digit passcode using the numbers 0-9, then there are 9⁴ options or 6,561 different codes to lock and unlock your phone.
To solve a permutation without repetition or where order matters, then the formula is:
where n is the number of options and r is the number of positions or slots to fill. The exclamation point (representing the factorial) means that the number is multiplied by each number below it until 1 is reached. In other words, 4! = 4 x 3 x 2 x 1. In the example above, if your phone does not allow you to repeat a number, then the formula is:
which equals 9 x 8 x 7 x 6 = 3,024. There are fewer choices because order matters.
To solve a combination, the formula is:
where n is the number of options and r is the number of positions or slots to fill. If there are 9 options for hamburger toppings and you have a coupon for 3 free toppings, how many different combinations can you order? Remember order does not matter because ketchup and mustard are the same as mustard and ketchup. The formula would be:
which simplifies to:
This equals 504 / 6 = 84. There are 84 topping combinations you can order.
Geometric probability can be measured when considering two shapes where one is inside the other. For example, if a dartboard is a circle with a 12-inch radius, and the bullseye has a 1-inch radius, what is the probability of hitting the bullseye? First, find the area of the two circles in terms of π, using the formula A = πr². The dartboard is 144π in² and the bullseye is 1π in². The probability of hitting a bullseye is: