Pack B — Fluency

Mathematics

Year 11 · Sport – Conditional Probability and Decision-Making

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–20

1.
Let $U = \{1,2,3,4,5,6,7,8\}$ , $A = \{ {setA} \}$ and $B = \{ {setB} \}$ . List the elements of $A \cap B$ .
2.
Using $A = \{ {setA} \}$ and $B = \{ {setB} \}$ , list $A \cup B$ .
3.
$U = \{1,2,3,4,5,6,7,8,9,10\}$ and $A = \{ {setA} \}$ . List $A'$ .
4.
In a Venn diagram, $n(A) = 15$ , $n(B) = 10$ and $n(A \cap B) = 6$ . Find $n(A \cup B)$ .
5.
In a class of 25 students, 14 play tennis, 10 play hockey, and 4 play both. How many play neither?
6.
Let $A = \{ {setA} \}$ . State whether $5 \in A$ or $5 \notin A$ .
7.
Let $A = \{2, 4, 6\}$ and $B = \{ {setB} \}$ . State whether $A \subseteq B$ is true or false.
8.
List the elements of $\{x \in \mathbb{Z} : -2 \leq x < 3\}$ .
9.
Let $A = \{ {setA} \}$ and $B = \{ {setB} \}$ . Find $A \cap B$ .
10.
In a universal set $U$ with $n(U) = 40$ , $n(A) = 15$ . Find $n(A')$ .
11.
A fair die is rolled. Find $P(\text{score is } a multiple of 3)$ .
12.
If $P(A) = \dfrac{2}{5}$ , find $P(A')$ .
13.
A bag contains 4 red and 6 blue counters. A counter is drawn at random. Find $P(\text{red})$ .
14.
A coin is tossed twice. Find $P(\text{two heads})$ .
15.
A spinner is spun 200 times. The result "blue" came up 75 times. Estimate $P(\text{ {label}})$ .
16.
A two-way table shows: 30 fish, 50 no fish; 50 meat, 30 no meat. Of 80 respondents, what is $P(\text{meat})$ ?
17.
Two fair dice are rolled. Find $P(\text{sum} = 7)$ .
18.
From a Venn diagram with $n(A) = 12$ , $n(B) = 9$ , $n(A \cap B) = 4$ , $n(U) = 25$ , find $P(A)$ .
19.
A card is drawn at random from a standard 52-card pack. Find $P(\text{ {desc}})$ .
20.
A spinner has outcomes Red, Blue, Green with $P(R) = 0.4$ and $P(B) = 0.35$ . Find $P(G)$ .

SilverQuestions 21–40

21.
In a survey of 40 people, 25 like tea, 20 like coffee, and 5 like neither. How many like both?
22.
In a class of 40, 22 study French, 15 study Spanish and 6 study both. How many study only French?
23.
$U = \{1, 2, \ldots, 15\}$ , $A = \{$ multiples of 3 $\}$ , $B = \{$ factors of 12 $\}$ . Find $A \cap B$ .
24.
$n(A) = 30$ , $n(B) = 24$ , $n(C) = 20$ . Pair intersections: $n(A\cap B) = 10$ , $n(A\cap C) = 8$ , $n(B\cap C) = 7$ . Triple: $n(A\cap B\cap C) = 4$ . Find $n(A \cup B \cup C)$ .
25.
Write the set $\{x \in \mathbb{R} : x > 3\}$ in interval notation.
26.
$U = \{1, 2, 3, 4, 5\}$ , $A = \{1, 3, 5\}$ , $B = \{2, 3\}$ . List $(A \cup B)'$ .
27.
Are $A = \{$ letters in MATHS $\}$ and $B = \{$ letters in STAMH $\}$ equal sets?
28.
Given $n(U) = 50$ , $n(A) = 30$ , $n(B) = 20$ and $n(A \cup B)' = 5$ , find $n(A \cap B)$ .
29.
Two fair dice are rolled. State the size of the sample space and list the outcomes giving a sum of 9.
30.
Let $A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$ . List $(A \cup B) \setminus (A \cap B)$ .
31.
Given $P(A) = 0.4$ , $P(B) = 0.5$ and $P(A \cap B) = 0.15$ , find $P(A \cup B)$ .
32.
A bag has 6 red and 3 blue counters. Two are drawn without replacement. Find $P(\text{both red})$ .
33.
A bag has 4 red and 3 blue counters. Two are drawn without replacement. Find $P(\text{one of each colour})$ .
34.
A two-way table shows: 9 pupils take French only, 6 take Spanish only, 6 take both, 9 take neither. A pupil is chosen at random. Given they take French, find $P(\text{Spanish})$ .
35.
Events $A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.6$ . Find $P(A \cap B)$ .
36.
Given $P(A \cap B) = 0.12$ and $P(A) = 0.4$ , find $P(B \mid A)$ .
37.
A card is drawn from a standard 52-card pack. Find $P(\text{heart or spade})$ and say whether the events are mutually exclusive.
38.
A spinner lands on red with probability 0.3. It is spun 4 times. Find $P(\text{at least one red})$ .
39.
Of 100 students, 60 own a phone, 40 own a tablet, 30 own both. A student who owns a tablet is chosen. Find $P(\text{also owns a phone})$ .
40.
$A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.3$ . Find $P(A \cup B)$ .

GoldQuestions 41–60

41.
In a class of 60, 30 study French, 24 study Spanish, 20 study German. 10 study French \& Spanish, 8 study French \& German, 7 study Spanish \& German, and 4 study all three. How many study at least one language?
42.
For the same class (use the totals above), how many study exactly one of the three languages?
43.
$U = \{1, 2, \ldots, 15\}$ , $A = \{$ multiples of 3 $\}$ , $B = \{$ factors of 12 $\}$ . Find $P(A' \cap B)$ when an element is chosen at random.
44.
In a 2-set Venn diagram for $A$ and $B$ , describe in words the region $A \cap B'$ , and give the number-elements interpretation.
45.
In a survey of $n(U) = 100$ students, $n(A) = 60$ and $n(B) = 45$ . If $n(A \cap B) = x$ and $n(A \cup B)' = y$ , write an equation relating $x$ and $y$ .
46.
In a Venn diagram with three sets $A$ , $B$ , $C$ , the region " $A$ only" has 12 students, " $B$ only" has 9, " $C$ only" has 7. Each pair-only region has 4 students. The all-three region has 3 students. There are 5 students in none of the three sets. Find $n(U)$ .
47.
For $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$ , $A = \{1, 2, 5, 6\}$ , $B = \{2, 3, 6, 7\}$ , find $A' \cap B$ .
48.
A coin and a 6-sided die are tossed together. Write the sample space $S$ as a set, and give $n(S)$ .
49.
Use De Morgan's laws to rewrite $(A \cup B)'$ without a union.
50.
In a 2-set Venn diagram, " $A$ only" has $2x$ students, " $B$ only" has $x + 5$ , "both" has $x$ , "neither" has $4$ . If $n(U) = 25$ , find $x$ .
51.
A factory has two machines: $M_1$ makes $70\%$ of items with defect rate $3\%$ , and $M_2$ makes $30\%$ with defect rate $6\%$ . An item is picked at random. Find $P(\text{defective})$ .
52.
Given $P(A) = 0.5$ , $P(B) = 0.3$ and $P(A \cap B) = 0.15$ , determine whether $A$ and $B$ are independent.
53.
Three fair coins are tossed. Find $P(\text{exactly {k} heads})$ .
54.
A box has 6 red and 4 blue balls. Three are drawn without replacement. Find $P(\text{all three red})$ .
55.
In a survey, $n(A) = 30$ , $n(B) = 25$ , $n(A \cap B) = 10$ and $n(U) = 60$ . Find $P(A \mid B)$ .
56.
Given $P(A \cup B) = 0.7$ , $P(B) = 0.4$ and $P(A \cap B) = 0.1$ , find $P(A)$ .
57.
$3\%$ of a population has a condition. A test is positive for $85\%$ of those who have it, and falsely positive for $4\%$ of those who do not. Find $P(\text{positive test})$ .
58.
At a school, $50\%$ study French and $30\%$ study Spanish. Of those who study French, $20\%$ also study Spanish. Find $P(F \cap S)$ .
59.
Using the factory in G1 (Pack A: $M_1$ 60%/2%, $M_2$ 40%/5%; Pack B: $M_1$ 70%/3%, $M_2$ 30%/6%), find $P(M_1 \mid \text{defective})$ .
60.
$A$ , $B$ , $C$ are mutually independent with $P(A) = P(B) = P(C) = p$ . Find $P(\text{exactly one of } A, B, C)$ when $p = \dfrac{1}{4}$ .

PlatinumQuestions 61–80

61.
Of 80 students, 40 like Maths, 35 like Science, 28 like Art; 15 like Maths \& Science, 10 like Maths \& Art, 8 like Science \& Art, and 4 like all three. How many like exactly one of the three subjects?
62.
Verify with $U = \{1,2,3,4,5,6\}$ , $A = \{1, 2, 3\}$ , $B = \{3, 4, 5\}$ that $(A \cup B)' = A' \cap B'$ .
63.
A class has 30 students. 18 study French, 16 study Spanish, and 4 study neither. Let $x$ = number studying both. Find $x$ , and the number studying exactly one of the two languages.
64.
A four-digit code is formed using the digits $1, 2, 3, 4, 5$ without repetition. (a) State the size of the sample space. (b) Let $E$ = "the code is even". Find $n(E)$ .
65.
In a class of 50 students, 30 take Maths (M), 20 take Physics (P), 25 take Chemistry (C). 10 take M and P, 8 take M and C, 5 take P and C, and 3 take all three. How many take **none** of the three?
66.
Express in set-builder notation, then in interval notation, the set of all real $x$ satisfying both $x \geq 2$ and $x < 7$ .
67.
Of 200 customers, 120 bought coffee, 80 bought a pastry, and 50 bought both. A customer is chosen at random from those who bought a pastry. (a) How many candidates are there? (b) Of those, how many also bought coffee?
68.
In a survey of 60 students, every student plays at least one of football (F), basketball (B), or tennis (T). 30 play F, 28 play B, 20 play T. 10 play F and B, 12 play F and T, 8 play B and T. If $x$ students play all three sports, find $x$ .
69.
List all subsets of $A = \{a, b, c\}$ . How many are there?
70.
A school newspaper survey says: "Of those who read the print edition, 60% also read the online edition." Let $P$ = "reads print" and $O$ = "reads online". Express the statement in set notation using $n(\cdot)$ , $\cap$ and $\cup$ .
71.
A diagnostic test: 2% of a population has a condition, $P(T^+ \mid C) = 0.9$ , $P(T^+ \mid C') = 0.05$ . Given a positive test, find $P(C \mid T^+)$ .
72.
In a town, $30\%$ commute by bike; of cyclists $20\%$ are late, of non-cyclists $5\%$ are late. Find $P(\text{cyclist} \mid \text{late})$ .
73.
A bag has 5 red, 2 blue and 3 green balls. Two are drawn without replacement. Find $P(\text{second is red} \mid \text{first is not red})$ .
74.
A shooter scores with probability $0.4$ . How many shots so that $P(\text{at least one score}) \geq 0.95$ ?
75.
A bag has 3 red, 2 blue and 1 green ball. Two are drawn without replacement. Find $P(\text{same colour})$ .
76.
$P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \mid B) = 0.6$ . Find $P(A \cup B)$ and decide whether $A$ and $B$ are independent.
77.
A vaccine has 70% chance of being effective per person. Find $P(\text{exactly 3 of 5 benefit})$ .
78.
A four-digit code is formed using digits $1,2,3,4,5$ without repetition. Find $P(\text{code is even})$ .
79.
Two fair dice are rolled. Given that the sum is even, find $P(\text{both dice show the same number})$ .
80.
In a group of 50, $n(A) = 30$ , $n(B) = 20$ and $n(A \cup B)' = 8$ . Find (a) $n(A \cap B)$ , (b) $P(A \mid B)$ .