Question

George and hamza both roll two standard fair $6$-sided dice. what is the probability that hamza rolls the same two numbers as george, though not necessarily in the same order?

Answer 1

The probability that Hamza rolls the same two numbers as George, though not necessarily in the same order, is 1/6.

Step-by-step explanation:

To find the probability that Hamza rolls the same two numbers as George, though not necessarily in the same order, we need to consider the number of possible outcomes.

When George rolls two standard fair 6-sided dice, there are 36 total outcomes in the sample space. This is because each of the two dice has 6 possible outcomes, so the total number of outcomes is 6*6 = 36.

Out of these 36 outcomes, there are 6 outcomes where Hamza rolls the same two numbers as George, but not necessarily in the same order. For example, if George rolls a 2 and a 4, Hamza could roll a 4 and a 2.

Therefore, the probability that Hamza rolls the same two numbers as George, though not necessarily in the same order, is 6/36 = 1/6.

Answer 2

The probability that Hamza rolls the same two numbers as George with two six-sided dice is 1/6, as there are always 6 outcomes out of 36 that would match George's roll, regardless of what he rolls.

Step-by-step explanation:

We want to calculate the probability that Hamza rolls the same two numbers as George with two six-sided dice. There are 36 (6x6) possible outcomes when rolling two dice. However, since the order does not matter, we need to consider combinations of the dice rolls rather than permutations. For instance, if George rolls a 3 and a 5, they could come in any order (3,5) or (5,3), but it is still the same combination for the purpose of matching Hamza's roll.

To calculate this, we first consider the probability that Hamza matches George's roll when George rolls a pair (same number on both dice). There are 6 cases like this (1,1), (2,2), ..., (6,6), and for each case, there is only one way Hamza can roll the same numbers. So for pairs, the probability that Hamza matches George is 1/36 for each.

For non-pair rolls, there are 6x5/2 = 15 unique non-pair combinations since order doesn't matter (1,2) and (2,1) are considered the same. For each of these combinations, there are 2 ways Hamza can match George's (1,2 or 2,1). Therefore, the probability for each non-pair combination that Hamza matches George is 2/36 or 1/18.

Add up the probabilities of matching pairs and non-pairs, weighted by the number of such occurrences:
(6*(1/36) + 15*(1/18)) = (1/6 + 5/6) = 1. Thus, the overall probability is 1/6. This is because no matter what George rolls, there are always 6 outcomes out of 36 that Hamza could possibly roll to match George's roll, making the probability 6/36, which simplifies to 1/6.