A small combination padlock can be opened by entering three different specified numbers from one to nine in a specified order. Benjamin has 10 attempts to guess the combination. Determine the probability of him opening the lock.
The probability of an event occurring equals the number of all ways that event can occur over all possible outcomes. In our case, we want to know the probability of Benjamin guessing correctly. Benjamin has 10 guesses. So there are 10 possible ways it could happen.
But now, we need to find out what all possible combinations on the lock there could be. We know that there are three different numbers, which means the numbers do not repeat themselves. And that’s helpful. If this is the padlock, the first digit could be any value from one to nine.
There are nine possible options for the first digit. And the second digit has to be one through nine minus one minus whatever the first digit was. And that means there’re only eight possible options for the second number. And then, the third digit can’t be the first or second digit, which means there are only seven possible options. The total number of options for the combination will be nine times eight times seven, which is equal to 504. The probability of Benjamin opening the lock is 10 guesses out of 504 possible combinations.
But when we work with probability, we wanna make sure we reduce this fraction as much as possible. Both the numerator and the denominator here are even numbers, which means they’re divisible by two. 10 divided by two equals five. 504 divided by two equals 252. Five and 252 do not share any common factors. We can say this because five is a prime number. And 252 is not divisible by five.
Since we can’t reduce this any further, we’ll say that the probability of Benjamin opening the lock is five out of 252.