Pack B — Fluency — 11.2 Probability

BronzeQuestions 1–10

1.
A fair die is rolled. Find $P(\text{score is } a multiple of 3)$ .
2.
If $P(A) = \dfrac{2}{5}$ , find $P(A')$ .
3.
A bag contains 4 red and 6 blue counters. A counter is drawn at random. Find $P(\text{red})$ .
4.
A coin is tossed twice. Find $P(\text{two heads})$ .
5.
A spinner is spun 200 times. The result "blue" came up 75 times. Estimate $P(\text{ {label}})$ .
6.
A two-way table shows: 30 fish, 50 no fish; 50 meat, 30 no meat. Of 80 respondents, what is $P(\text{meat})$ ?
7.
Two fair dice are rolled. Find $P(\text{sum} = 7)$ .
8.
From a Venn diagram with $n(A) = 12$ , $n(B) = 9$ , $n(A \cap B) = 4$ , $n(U) = 25$ , find $P(A)$ .
9.
A card is drawn at random from a standard 52-card pack. Find $P(\text{ {desc}})$ .
10.
A spinner has outcomes Red, Blue, Green with $P(R) = 0.4$ and $P(B) = 0.35$ . Find $P(G)$ .

SilverQuestions 11–20

11.
Given $P(A) = 0.4$ , $P(B) = 0.5$ and $P(A \cap B) = 0.15$ , find $P(A \cup B)$ .
12.
A bag has 6 red and 3 blue counters. Two are drawn without replacement. Find $P(\text{both red})$ .
13.
A bag has 4 red and 3 blue counters. Two are drawn without replacement. Find $P(\text{one of each colour})$ .
14.
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 $P(\text{Spanish})$ .
15.
Events $A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.6$ . Find $P(A \cap B)$ .
16.
Given $P(A \cap B) = 0.12$ and $P(A) = 0.4$ , find $P(B \mid A)$ .
17.
A card is drawn from a standard 52-card pack. Find $P(\text{heart or spade})$ and say whether the events are mutually exclusive.
18.
A spinner lands on red with probability 0.3. It is spun 4 times. Find $P(\text{at least one red})$ .
19.
Of 100 students, 60 own a phone, 40 own a tablet, 30 own both. A student who owns a tablet is chosen. Find $P(\text{also owns a phone})$ .
20.
$A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.3$ . Find $P(A \cup B)$ .

GoldQuestions 21–30

21.
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 $P(\text{defective})$ .
22.
Given $P(A) = 0.5$ , $P(B) = 0.3$ and $P(A \cap B) = 0.15$ , determine whether $A$ and $B$ are independent.
23.
Three fair coins are tossed. Find $P(\text{exactly {k} heads})$ .
24.
A box has 6 red and 4 blue balls. Three are drawn without replacement. Find $P(\text{all three red})$ .
25.
In a survey, $n(A) = 30$ , $n(B) = 25$ , $n(A \cap B) = 10$ and $n(U) = 60$ . Find $P(A \mid B)$ .
26.
Given $P(A \cup B) = 0.7$ , $P(B) = 0.4$ and $P(A \cap B) = 0.1$ , find $P(A)$ .
27.
$3\%$ 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 $P(\text{positive test})$ .
28.
At a school, $50\%$ study French and $30\%$ study Spanish. Of those who study French, $20\%$ also study Spanish. Find $P(F \cap S)$ .
29.
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 $P(M_1 \mid \text{defective})$ .
30.
$A$ , $B$ , $C$ 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 31–40

31.
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 $P(C \mid T^+)$ .
32.
In a town, $30\%$ commute by bike; of cyclists $20\%$ are late, of non-cyclists $5\%$ are late. Find $P(\text{cyclist} \mid \text{late})$ .
33.
A bag has 5 red, 2 blue and 3 green balls. Two are drawn without replacement. Find $P(\text{second is red} \mid \text{first is not red})$ .
34.
A shooter scores with probability $0.4$ . How many shots so that $P(\text{at least one score}) \geq 0.95$ ?
35.
A bag has 3 red, 2 blue and 1 green ball. Two are drawn without replacement. Find $P(\text{same colour})$ .
36.
$P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \mid B) = 0.6$ . Find $P(A \cup B)$ and decide whether $A$ and $B$ are independent.
37.
A vaccine has 70% chance of being effective per person. Find $P(\text{exactly 3 of 5 benefit})$ .
38.
A four-digit code is formed using digits $1,2,3,4,5$ without repetition. Find $P(\text{code is even})$ .
39.
Two fair dice are rolled. Given that the sum is even, find $P(\text{both dice show the same number})$ .
40.
In a group of 50, $n(A) = 30$ , $n(B) = 20$ and $n(A \cup B)' = 8$ . Find (a) $n(A \cap B)$ , (b) $P(A \mid B)$ .

Pack B — Fluency