CalcGospel 國際數學圖譜
返回

IAL 2025 Jan Q3

A Level / Edexcel / S1

IAL 2025 Jan Paper · Question 3

题目

Problem

A factory makes rugs and blankets from wool or cotton or polyester.

  • 30%30\% of the items made are rugs and 70%70\% of the items made are blankets
  • 60%60\% of the rugs are made from wool
  • 35%35\% of the blankets are made from wool
  • 10%10\% of the rugs are made from cotton
  • 20%20\% of the blankets are made from cotton
  • the rest of the rugs and blankets are made from polyester

An item made by the factory is selected at random.

(a) Complete the tree diagram below to illustrate this information.

Tree diagram
(2)

(b) (i) Find the probability that the item selected is not made from wool.

(2)

(ii) Given that this item is not made from wool, find the probability that it is a blanket.

(2)

解答

(a)

解法一

思路

展开

第一层分 rug 和 blanket，概率分别是 0.30.3 和 0.70.7。第二层按材料分，polyester 是剩下的概率。

答题过程

展开

The first branches are

P(rug)=0.3,P(blanket)=0.7.\begin{align*} P(\text{rug})=0.3,\qquad P(\text{blanket})=0.7. \end{align*}

For rugs,

P(wool∣rug)= 0.6,P(cotton∣rug)= 0.1,P(polyester∣rug)= 1−0.6−0.1=0.3.\begin{align*} P(\text{wool}\mid\text{rug})=&\,0.6,\\[2mm] P(\text{cotton}\mid\text{rug})=&\,0.1,\\[2mm] P(\text{polyester}\mid\text{rug})=&\,1-0.6-0.1=0.3. \end{align*}

For blankets,

P(wool∣blanket)= 0.35,P(cotton∣blanket)= 0.20,P(polyester∣blanket)= 1−0.35−0.20=0.45.\begin{align*} P(\text{wool}\mid\text{blanket})=&\,0.35,\\[2mm] P(\text{cotton}\mid\text{blanket})=&\,0.20,\\[2mm] P(\text{polyester}\mid\text{blanket})=&\,1-0.35-0.20=0.45. \end{align*}

(b)(i)

解法一

思路

展开

“Not wool” 可以来自 rug 且不是 wool，也可以来自 blanket 且不是 wool。沿 tree diagram 两条大分支分别乘，再相加。

答题过程

展开 P(W′)= P(rug and not wool)+P(blanket and not wool)= 0.3(0.1+0.3)+0.7(0.2+0.45)= 0.3(0.4)+0.7(0.65)= 0.12+0.455= 0.575.\begin{align*} P(W') =&\,P(\text{rug and not wool}) +P(\text{blanket and not wool})\\[3mm] =&\,0.3(0.1+0.3)+0.7(0.2+0.45)\\[3mm] =&\,0.3(0.4)+0.7(0.65)\\[3mm] =&\,0.12+0.455\\[3mm] =&\,0.575. \end{align*}

解法二

思路

展开

对立事件法。直接计算选中产品是由羊毛（wool）制成的概率 P(W)P(W)，然后利用对立事件公式 P(W′)=1−P(W)P(W') = 1 - P(W) 来计算“不是由羊毛制成”的概率。这种方法在求包含“不是/不满足”某条件的概率时，通常计算量较小且逻辑清晰。

答题过程

展开

First calculate the probability that the selected item is made from wool:

P(W)=  P(rug and wool)+P(blanket and wool)=  0.3(0.6)+0.7(0.35)=  0.18+0.245=  0.425\begin{align*} P(W) =&\,\, P(\text{rug and wool}) + P(\text{blanket and wool})\\[3mm] =&\,\, 0.3(0.6) + 0.7(0.35)\\[3mm] =&\,\, 0.18 + 0.245\\[3mm] =&\,\, 0.425 \end{align*}

Then calculate the complementary probability:

P(W′)=  1−P(W)=  1−0.425=  0.575.\begin{align*} P(W') =&\,\, 1 - P(W)\\[3mm] =&\,\, 1 - 0.425\\[3mm] =&\,\, 0.575. \end{align*}

(b)(ii)

解法一

思路

展开

这是条件概率：

P(blanket∣W′)=P(blanket and not wool)P(W′).P(\text{blanket}\mid W') =\frac{P(\text{blanket and not wool})}{P(W')}.

答题过程

展开 P(blanket∣W′)= P(blanket and not wool)P(W′)= 0.7(0.2+0.45)0.575= 0.4550.575= 91115.\begin{align*} P(\text{blanket}\mid W') =&\,\frac{P(\text{blanket and not wool})}{P(W')}\\[3mm] =&\,\frac{0.7(0.2+0.45)}{0.575}\\[3mm] =&\,\frac{0.455}{0.575}\\[3mm] =&\,\frac{91}{115}. \end{align*}