### Instruction: Jack rolls 5 fair six-sided dice. What is the probability that at least two dice show the same number?### Response: This is a probability problem, so I need to find the favorable outcomes and divide them by the total possible outcomes.
The total possible outcomes are easy to calculate: since each die has 6 options, and there are 5 dice, there are 6^5 = 7776 ways to roll the dice.
The favorable outcomes are a bit trickier: I need to count how many ways there are to have at least two dice show the same number.
One way to approach this is to use the complement rule: the probability of at least two dice matching is equal to one minus the probability of no dice matching.
So, how can I find the probability of no dice matching?
Well, that means each die has to show a different number from the others.
The first die can show any number, so it has 6 options.
The second die can show any number except the one on the first die, so it has 5 options.
The third die can show any number except the ones on the first and second dice, so it has 4 options.
And so on.
Multiplying the options for each die, I get 6 x 5 x 4 x 3 x 2 = 720 ways to have no dice matching.
Dividing this by the total possible outcomes, I get 720/7776 = 0.0926 as the probability of no dice matching.
Using the complement rule, I subtract this from 1 to get the probability of at least two dice matching: 1 - 0.0926 = 0.9074.