题目

Problem

A group of $200$ adults were asked whether they read cooking magazines, travel magazines or sport magazines.

Their replies showed that

$29$ read only cooking magazines
$33$ read only travel magazines
$42$ read only sport magazines
$17$ read cooking magazines and sport magazines but not travel magazines
$11$ read travel magazines and sport magazines but not cooking magazines
$22$ read cooking magazines and travel magazines but not sport magazines
$32$ do not read cooking magazines, travel magazines or sport magazines

(a) Using this information, complete the Venn diagram on page 11.

Venn diagram for Cooking, Travel and Sport magazines

(3)

One of these adults was chosen at random.

(b) Find the probability that this adult,

(i) reads cooking magazines and travel magazines and sport magazines,

(1)

(ii) does not read cooking magazines.

(2)

Given that this adult reads travel magazines,

(2)

解答

(a)

解法一

思路

展开

题目已经给出所有“only”和“两类但不是第三类”的人数，以及三个都不读的人数。唯一缺的是三类都读的人数。

把已知人数相加，再用总人数 $200$ 减去即可。

答题过程

展开

The number who read all three types of magazine is

\begin{align*} 200&-(29+33+42+17+11+22+32)\\[3mm] =&\,200-186\\[3mm] =&\,14. \end{align*}

So the completed Venn diagram has:

\begin{array}{c|c} \text{Region} & \text{Frequency}\\ \hline \text{Cooking only} & 29\\ \text{Travel only} & 33\\ \text{Sport only} & 42\\ \text{Cooking and Sport only} & 17\\ \text{Travel and Sport only} & 11\\ \text{Cooking and Travel only} & 22\\ \text{All three} & 14\\ \text{None} & 32 \end{array}

(b)(i)

解法一

思路

展开

三类都读的人数是上一小题求出的 $14$ ，总人数是 $200$ 。

答题过程

展开

P(\text{Cooking and Travel and Sport}) =\frac{14}{200}.

(b)(ii)

解法一

思路

展开

不读 cooking 的区域包括：只读 travel、只读 sport、travel and sport only，以及三类都不读。

答题过程

展开

The number who do not read cooking magazines is

\begin{align*} 33+42+11+32=118. \end{align*}

Therefore

P(\text{does not read cooking}) =\frac{118}{200}.

解法二

思路

展开

对立事件法。要求“不读烹饪杂志（cooking）”的人数，可以用总人数 $200$ 减去“读烹饪杂志”的总人数。读烹饪杂志的人数可以通过文氏图中属于 Cooking 圆圈的四个区域相加得到。这种方法可以有效避免在正面数数时漏掉某些不读烹饪的区域。

答题过程

展开

First calculate the total number of adults who read cooking magazines:

\begin{align*} n(\text{Cooking}) =&\,\, 29 + 22 + 17 + 14\\[3mm] =&\,\, 82. \end{align*}

Then the number of adults who do not read cooking magazines is:

\begin{align*} n(\text{Cooking}') =&\,\, 200 - n(\text{Cooking})\\[3mm] =&\,\, 200 - 82\\[3mm] =&\,\, 118. \end{align*}

Therefore, the probability is:

\begin{align*} P(\text{does not read cooking}) = \frac{118}{200}. \end{align*}

(c)

解法一

思路

展开

这是条件概率。已知这个人读 travel，所以分母是所有读 travel 的人数。分子是同时读 travel 和 sport 的人数。

答题过程

展开

The number who read travel magazines is

\begin{align*} 33+22+11+14=80. \end{align*}

The number who read both travel and sport magazines is

\begin{align*} 11+14=25. \end{align*}

Therefore

P(\text{Sport}\mid \text{Travel}) =\frac{25}{80}.

IAL 2024 Oct Q3

题目

解答

(a)

解法一

思路

答题过程

(b)(i)

解法一

思路

答题过程

(b)(ii)

解法一

思路

答题过程

解法二

思路

答题过程

(c)

解法一

思路

答题过程