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1. There are $9$ sweets in a bag: $3$ red, $2$ blue, and $4$ yellow. If I take out two sweets at random, find the probability I take a red and a yellow sweet:
  1. with replacement;
  2. without replacement.
2. $100$ people were surveyed.
$65$ people like running, $48$ people like swimming and $60$ people like cycling. $40$ people like running and swimming, $30$ people like swimming and cycling, and $35$ people like running and cycling. $25$ people like all three.
Given that a randomly selected person likes running, what is the probability that they like swimming and do not like cycling?
3. Lexie is late to school with probability $0.5$ when the weather is good, and $0.8$ when the weather is bad.
There is a $60$% chance the weather is good on any given day.
Given that Lexie was late to school today, what is the probability that the weather was good?
4. $100$ students were asked if they liked English, Maths, and/or Science.
$90$ like English, $90$ like Maths, $92$ like Science, $88$ like English and Maths, $86$ like Maths and Science, $87$ like English and Science, $85$ like all three.
  1. Find the probability that a random student likes exactly two subjects.
  2. A student who likes at least two subjects is selected. Find the probability that they like English.
5. Leyla misses maths lessons with probability $0.1$.
When she misses a maths lesson, the probability that she does the work is $0.6$.
When she does not miss a lesson, the probability that she does the work is $0.9$.
Given that Leyla has done the work, what is the probability that she missed the maths lesson?
6. Two events $A$ and $B$ are such that $$\mathrm{P}(A) = 0.3 \quad \mathrm{P}(B) = 0.5 \quad \mathrm{P}(A \cup B) = 0.6$$ Find:
  1. $\mathrm{P}(A \cap B)$
  2. $\mathrm{P}(A' \cup B)$
7. A teacher plays a maths game with one of their students. The probability that the teacher wins is $0.8$. They play three games.
  1. Find the probability that the teacher wins all three.
  2. Comment on any assumption(s) you have made and their validity.
8. Two events $A$ and $B$ are such that $$\mathrm{P}(A) = 0.3 \quad \mathrm{P}(A\cap B') = 0.1 \quad \mathrm{P}(A\cup B') = 0.55$$ Find $\mathrm{P}(B'|A')$.
9. Two events $A$ and $B$ are such that $$\mathrm{P}(B) = 0.76 \quad \mathrm{P}(B|A) = 0.6 \quad \mathrm{P}(A'\cap B') = 0$$ Find $\mathrm{P}(A)$.
10. Two events $A$ and $B$ are independent, with $$\mathrm{P}(A) = 0.3 \quad \mathrm{P}(B) = 0.5$$ Find:
  1. $\mathrm{P}(A' \cap B)$
  2. $\mathrm{P}(B|A')$
11. Two events $A$ and $B$ are such that $$\mathrm{P}(A) = 0.45 \quad \mathrm{P}(A \cap B') = 0.25 \quad \mathrm{P}(A \cup B) = 0.8$$ Find:
  1. $\mathrm{P}(B'|A')$
  2. $\mathrm{P}(A \cap B'|A'\cup B')$
12. Two events $A$ and $B$ are such that $$\mathrm{P}(A|B) = 1 \quad \mathrm{P}(A|B') = 0.25 \quad \mathrm{P}(B) = 0.6$$ Find $\mathrm{P}(B'|A)$.
13. Two events $A$ and $B$ are such that $$\mathrm{P}(A) = 0.6 \quad \mathrm{P}(B|A) = \frac{1}{6} \quad \mathrm{P}(A|B) = \frac{3}{13}$$ Find $\mathrm{P}(A'\cap B')$.
14. Two events $A$ and $B$ are such that $$\mathrm{P}(B|A) = \frac{3}{8} \quad \mathrm{P}(A|B) = \frac{4}{9} \quad \mathrm{P}(B|A') = \frac{15}{28}$$ Find $\mathrm{P}(A)$.
15. Two events $A$ and $B$ are such that $$\mathrm{P}(A) = 0.4 \quad \mathrm{P}(A|B) = 0.6 \quad \mathrm{P}(A\cup B) = 2\mathrm{P}(A\cap B)$$ Find $\mathrm{P}(B|A')$.
16. Amy and Ben are sometimes late to school. The probability Amy is late is $0.25$, the probability that both are late is $0.15$ and the probability that neither are late is $0.7$.
  1. Find the probability that Ben is late on any given day.
  2. Given that Amy is late one day, what is the probability that Ben is also late?
  3. Are Amy and Ben being late to school independent of each other?
17. There are 3 sweets, $A$, $B$ and $C$. Of $30$ people surveyed, $10$ like none of them, $6$ like $A$, $10$ like $B$, $9$ like $C$, $2$ like $A$ and $B$, $3$ like $B$ and $C$, and no one likes all three.
  1. Find the number of people who like both $A$ and $C$.
  2. Find the probability a random person selected likes more than $1$ sweet.
  3. Determine whether liking $B$ and liking $C$ are statistically independent.
18. Given $\mathrm{P}(A)=a$ and $\mathrm{P}(B)=b$, write $\mathrm{P}(A\cup B)$ in terms of $a$ and $b$ when:
  1. $A$ and $B$ are mutually exclusive;
  2. $A$ and $B$ are independent.