Final answer:
To find the probability of not having a pair of shoes that go together when opening four boxes, we multiply the number of favorable outcomes and divide by the total number of possible outcomes. The probability is 1/6. To find the probability of having at most two pairs when opening five boxes, we count the favorable outcomes for 0 pairs, 1 pair, and 2 pairs, and divide by the total number of possible outcomes. The probability is 7/20.
Step-by-step explanation:
To solve this problem, we need to use the concept of combinations. Let's start with part (a) of the question.
(a) The total number of ways to open four boxes in random order is 4!. Now, let's count the favorable outcomes where you do not have a pair that go together. The first box can be any of the four, the second box can be any of the three remaining, then the third box can be any of the two remaining, and finally, the fourth box is determined. So, the favorable outcomes are 4 * 3 * 2 * 1 = 24.
Therefore, the probability of not having a pair that go together is 24/4! = 1/6.
(b) To find the probability of having at most two pairs when opening five boxes, we need to count the favorable outcomes where we have 0 pairs, 1 pair, or 2 pairs. The total number of ways to open five boxes in random order is 5!. The favorable outcomes for 0 pairs is the same as part (a) - 24. The favorable outcomes for 1 pair is when the boxes are arranged in the pattern of 3+1+1 or 2+2+1, which gives us 4 * (3!/2!1!) or 3 * (4!/2!2!) outcomes, respectively. Lastly, the favorable outcomes for 2 pairs is when the boxes are arranged in the pattern of 2+2+1 or 2+1+1+1, which gives us 3 * (4!/2!2!) or 4 * (4!/3!1!) outcomes, respectively.
Therefore, the probability of having at most two pairs is (24 + 4 * 3 + 3 * 6)/5! = 7/20.