Just for your interest, here are a couple of questions from Japanese Senior High School entrance exam. [Yes, they do take entrance exams to get to Senior high in Japan. Their compulsory education end at the end of 9th grade, which coincides with the end of their junior high school. So, students taking this exam are 9th graders, going into 10th... if they pass the exam.]
[Please excuse rather rough translation.]
1 We are going to package a certain item. We will place a predetermined number of items in each small box, and when we have the same number of small boxes as the number of items that can be placed in the small box, we create a large box with that many small packages. The example below (actually shown with pictures) shows when the number used is 6. It shows one large box of 6 small boxes (all filled up), 1 small box with 6 items, and 5 loose items that could not be made into a small box package. Answer the following questions. Assume the number of items per small box is greater than or equal to 2.
a) Suppose we package 5 items per small box. There are 3 large boxes, 2 small boxes, and 3 loose items. How many items were there altogether?
b) You have 198 items. Find the number of items in a small box if you were to have 2 large boxes, 4 small boxes, with no left over items.
2 There are two switches that change the colors of lights. Switch A will alternate Bule and Red lights while Switch B will alternate Blue, Yellow, and Red lights. [these are two different circuits] Each switch will flash only one color each time the switch is engaged, and they alternate the colors in the orders described above. An experiment consists of rolling two dice when both lights are blue. You will push Switch A the number of times shown on the larger die, and push Swith B the number of times shown on the smaller dice and record the color of two lights. For example, if you roll 2 on the large die and 4 on the small die, you will hit Switch A twice and Switch B four times,
A: Blue --> Red --> Blue B: Blue-->Yellow-->Red-->Blue-->Yellow
Thus, at the end, Swith A will be showing blue light while Switch B will be showing yellow.
Find the following probability (reminder, we are always starting with both lights being blue)
a) Find the probability that both lights will be blue
b) Find the probability that the two lights show different colors.
Tad Watanabe Towson State University Towson, MD 21204