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: - with replacement;
- without replacement.
- $\dfrac{8}{27}$
- $\dfrac{1}{3}$
$100$ 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?
$\dfrac{3}{13}$
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?
$\dfrac{15}{31} = 0.484$
$100$ 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. - Find the probability that a random student likes exactly two subjects.
- A student who likes at least two subjects is selected. Find the probability that they like English.
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?
$\dfrac{2}{29} = 0.0690$
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: - $\mathrm{P}(A \cap B)$
- $\mathrm{P}(A' \cup B)$
A teacher plays a maths game with one of their students. The probability that the teacher wins is $0.8$. They play three games. - Find the probability that the teacher wins all three.
- Comment on any assumption(s) you have made and their validity.
- $0.8^3 = 512$
- Assume the probability of the teacher winning is the same every game. This may not be true since the teacher may get tired or lose motivation.
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 $\mathrm{P}(B'|A')$.
$\dfrac{5}{14}$
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 $\mathrm{P}(A)$.
$0.6$
Two events $A$ and $B$ are independent, with $$\mathrm{P}(A) = 0.3 \quad \mathrm{P}(B) = 0.5$$ Find: - $\mathrm{P}(A' \cap B)$
- $\mathrm{P}(B|A')$
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: - $\mathrm{P}(B'|A')$
- $\mathrm{P}(A \cap B'|A'\cup B')$
- $\dfrac{4}{11}$
- $\dfrac{5}{16}$
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 $\mathrm{P}(B'|A)$.
$\dfrac{1}{7}$
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 $\mathrm{P}(A'\cap B')$.
$\dfrac{1}{15}$
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 $\mathrm{P}(A)$.
$\dfrac{8}{15}$
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 $\mathrm{P}(B|A')$.
$\dfrac{1}{3}$
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 $0.7$. - Find the probability that Ben is late on any given day.
- Given that Amy is late one day, what is the probability that Ben is also late?
- Are Amy and Ben being late to school independent of each other?
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. - Find the number of people who like both $A$ and $C$.
- Find the probability a random person selected likes more than $1$ sweet.
- Determine whether liking $B$ and liking $C$ are statistically independent.
- $0$
- $\dfrac{1}{6}$
- Independent
Given $\mathrm{P}(A)=a$ and $\mathrm{P}(B)=b$, write $\mathrm{P}(A\cup B)$ in terms of $a$ and $b$ when: - $A$ and $B$ are mutually exclusive;
- $A$ and $B$ are independent.