MQ logo
1. $X$ is a discrete random variable with probability distribution:
$x$$1$$2$$3$$4$$5$$6$
$\mathrm{P}(X=x)$$\frac{1}{12}$$\frac{1}{6}$$\frac{1}{12}$$\frac{1}{4}$$2k$$3k$
where $k$ is a constant. Find $\mathrm{P}(X\leqslant5)$
2. A discrete probability distribution has $$\mathrm{P}(X\leqslant x) = kx^2$$ for $x=1,2,3$. Find the probability distribution for $X$.
3. Given that $\mathrm{P}(X=x) = \dfrac{1}{9}$ for integers in the range $-k\leqslant x \leqslant k$, find $k$.
4. A drawer contains $2$ black and $3$ white socks. Two socks are taken without replacement.
Find the probability distribution of $X$, the number of white socks taken.
5. The number of times Willow falls at any climbing competition, $X$, is given by $$\mathrm{P}(X=x) = k(4-x)$$ for $x = 0, 1, 2, 3$.
Find the probability that Willow falls at least once in a competition.
6. Given $X\sim B(30,0.3)$, calculate:
  1. $\mathrm{P}(X < 12)$
  2. $\mathrm{P}(X > 12)$
  3. $\mathrm{P}(7 < X \leqslant 14)$
7. Louise and Athina race to see who can do the most maths questions in a lesson. The number of questions Louise does, $X$, and the number of questions Athina does, $Y$, are given by the following distributions:
$n$123
$\mathrm{P}(X=n)$$0.2$$0.3$$0.5$
$\mathrm{P}(Y=n)$$0.1$$0.25$$0.65$
Find the probability that Louise does more questions than Athina.
8. The probability that Stella dislikes a teacher is $0.6$.
  1. This year, Stella has $10$ teachers. Find the probability that she dislikes fewer than 4 of them.
  2. Since she started at the school, she has had $40$ different teachers. Find the probability that she has disliked between $25$ and $30$ (inclusive) of her teachers.
  3. What are the limitations of the model you have used?
9. The probability of Max doing their homework on time is $0.6$.
  1. Find the probability that Max does three or more of their next seven homeworks on time.
  2. Given that Max did three or more of their last seven homeworks on time, what is the probability that they did exactly three on time?
10. Two independent observations of a random variable $X$ are made. Given $X\sim B(13,0.16)$, find the probability that exactly one of the observations is equal to $2$.
11. $X\sim B(15,p)$
  1. Given that $p = 0.4$, find $\mathrm{P}(X > 11)$
  2. Given instead that $\mathrm{P}(X = 0) = 0.03$, find $p$ correct to three decimal places.
12. $X\sim B(n,p)$. Given that the mean and standard deviation of $X$ are both equal to $0.9$, find $n$ and $p$.
13. $X\sim B(n,0.2)$, and the mean is three times as large as the standard deviation. Find $n$.
14. The chance of a sweet being lemon flavoured is $10$%. Erin buys $20$ sweets.
  1. Write down two conditions, in context, which must apply for a Binomial model to be valid for this situation.
  2. What is the probability that there is at least $1$ lemon flavoured sweet?
  3. Erin buys $n$ sweets instead such that there is at least a $95$% chance of there being a lemon flavoured sweet. Find the smallest value of $n$ for which this is true.
15. A school knows that $15$% of its students will fail to turn up to a school trip. The school books a coach which has space for $30$ students.
  1. Find the probability that the coach will be full if the school invites $30$ students onto the trip.
  2. The school accidentally invites $33$ students onto the trip. Find the probability that the school will be able to take all the students who turn up onto the trip.
16. The probability of Jamie getting a detention on any given week is $0.1$.
  1. In a term of $14$ weeks, find the probability Jamie receives more than $3$ detentions.
  2. After how many weeks will the average number of detentions Jamie has received exceed $5$?
  3. Students are expelled for receiving more than $5$ detentions in a year of $40$ weeks. Given that Jamie was not expelled, find the probability that they received $4$ detentions.
17. Dylan flips a coin $1000$ times to see if it is biased. It lands on heads $510$ times. Dylan claims that the coin must be biased because the probability of this happening is very low.
  1. Find the probability of an unbiased coin landing on heads $510$ times out of $1000$.
  2. Explain whether you agree with Dylan's claim.
18. A bag contains $3$ apple and $2$ banana flavoured sweets. A second bag contains $4$ apple and $3$ banana flavoured sweets. A sweet is selected at random from the first bag, then placed in the second bag.
A sweet is then selected from the second bag.
  1. Find the distribution of $X$, the number of apple flavoured sweets selected.
  2. This is repeated three times to get three different values of $X$. Let $Y$ be the median of these three values. Find the distribution of $Y$.