题目
A multiple-choice test consists of 25 questions, each having 5 responses, only one of which is correct.
Each correct answer gains 4 marks but each incorrect answer loses 1 mark.
Sam answers all 25 questions by choosing at random one response for each question.
Let be the number of correct answers that Sam achieves.
(a) State the distribution of
Let be the number of marks that Sam achieves.
(b) (i) State the distribution of in terms of
(ii) Hence, show clearly that the number of marks that Sam is expected to achieve is zero.
In order to pass the test at least 30 marks are required.
(c) Find the probability that Sam will pass the test.
Past records show that when the test is done properly, the probability that a student answers the first question correctly is 0.5
A random sample of 50 students that did the test properly was taken.
Given that the probability that more than but at most 30 students answered the first question correctly was 0.9328 to 4 decimal places,
(d) find the value of
(Total for Question 2 is 12 marks)
题目中文翻译
一项多项选择测试包含 25 道题，每题有 5 个选项，只有 1 个正确。
每答对一题得 4 分，每答错一题扣 1 分。
Sam 为每道题随机选择一个选项来回答所有 25 道题。
令 为 Sam 答对的题数。
(a) 说明 的分布。
令 为 Sam 获得的分数。
(b) (i) 用 表示 的分布。 (ii) 由此清楚地证明 Sam 预期获得的分数为零。
为了通过测试，至少需要 30 分。
(c) 求 Sam 通过测试的概率。
过往记录显示，当测试正常进行时，学生答对第一题的概率为 0.5。
随机抽取了 50 名正常参加测试的学生样本。
已知答对第一题的学生人数超过 但不超过 30 的概率为 0.9328（精确到 4 位小数），
(d) 求 的值。
（第 2 题共 12 分）