Question

name____ib applications year 1 graded classwork-probability (29 marks)in an international competition, participants can answer questions in only one of the three following languages: portuguese, mandarin or hindi. 80 participants took part in the competition. the number of participants answering in portuguese, mandarin or hindi is shown in the table.| participants | portuguese | mandarin | hindi | total || ---- | ---- | ---- | ---- | ---- || boys | 20 | 18 | 5 | 43 || girls | 18 | 7 | 12 | 37 || total | 38 | 25 | 17 | 80 |(a) state the number of boys who answered questions in portuguese. 1a boy is chosen at random.(b) find the probability that the boy answered questions in hindi. 2two girls are selected at random.(c) calculate the probability that one girl answered questions in mandarin and the other answered questions in hindi. 3emily starts reading leo tolstoys war and peace on the 1st of february. the number of pages she reads each day increases by the same number on each successive day.| date in february | 1 | 2 | 3 | 4 | ......... || ---- | ---- | ---- | ---- | ---- | ---- || number of pages read | 8 | 11 | 14 | 17 | ......... |(a) calculate the number of pages emily reads on the 14th of february. 3(b) find the exact total number of pages emily reads in the 28 days of february. 3

Explanation:

First Section (Probability)

Step1: Extract boys' Portuguese count

From the table, boys who answered in Portuguese: $20$

Step2: Calculate boy's Hindi probability

Total boys = $43$, Hindi boys = $5$. Probability = $\frac{\text{Hindi boys}}{\text{Total boys}}$
$\frac{5}{43}$

Step3: Calculate girl selection probability

Total girls = $37$. Mandarin girls = $7$, Hindi girls = $12$. Number of ways to choose 1 Mandarin + 1 Hindi girl: $7 \times 12$. Total ways to choose 2 girls: $\binom{37}{2}=\frac{37 \times 36}{2}=666$. Probability = $\frac{7 \times 12}{666}$
Simplify: $\frac{84}{666}=\frac{14}{111}$

Step1: Identify sequence parameters

This is an arithmetic sequence with first term $a_1=8$, common difference $d=11-8=3$.

Step2: Find 14th term

Use arithmetic sequence formula: $a_n = a_1 + (n-1)d$. For $n=14$:
$a_{14}=8 + (14-1)\times3 = 8 + 39 = 47$

Step3: Find total pages in 28 days

Use sum formula for arithmetic sequence: $S_n = \frac{n}{2}[2a_1 + (n-1)d]$. For $n=28$:
$S_{28}=\frac{28}{2}[2\times8 + (28-1)\times3] = 14[16 + 81] = 14\times97 = 1358$

Answer:

(a) $20$
(b) $\frac{5}{43}$
(c) $\frac{14}{111}$

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Second Section (Arithmetic Sequence)