$X$ 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 $\mathrm{P}(X\leqslant5)$ $k = \frac{1}{12}$ and $\mathrm{P}(X\leqslant5) = \frac{3}{4}$
A discrete probability distribution has $$\mathrm{P}(X\leqslant x) = kx^2$$ for $x=1,2,3$. Find the probability distribution for $X$.
<table><tr><td>$x$</td><td>$1$</td><td>$2$</td><td>$3$</td></tr><tr><td>$\mathrm{P}(X=x)$</td><td>$\frac{1}{9}$</td><td>$\frac{1}{3}$</td><td>$\frac{5}{9}$</td></tr></table>
Given that $\mathrm{P}(X=x) = \dfrac{1}{9}$ for integers in the range $-k\leqslant x \leqslant k$, find $k$.
There must be 9 integers, so $k=4$
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.
<table><tr><td>$x$</td><td>0</td><td>1</td><td>2</td></tr><tr><td>$\mathrm{P}(X=x)$</td><td>$\frac{1}{10}$</td><td>$\frac{6}{10}$</td><td>$\frac{3}{10}$</td></tr></table>
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.
$0.6$
Given $X\sim B(30,0.3)$, calculate: - $\mathrm{P}(X < 12)$
- $\mathrm{P}(X > 12)$
- $\mathrm{P}(7 < X \leqslant 14)$
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: | $n$ | 1 | 2 | 3 |
| $\mathrm{P}(X=n)$ | $0.2$ | $0.3$ | $0.5$ |
| $\mathrm{P}(Y=n)$ | $0.1$ | $0.25$ | $0.65$ |
Find the probability that Louise does more questions than Athina. $0.205$
The probability that Stella dislikes a teacher is $0.6$. - This year, Stella has $10$ teachers. Find the probability that she dislikes fewer than 4 of them.
- 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.
- What are the limitations of the model you have used?
- $0.0548$
- $0.425$
- The probability Stella likes her teacher is independent of her liking another teacher, and the probability Stella likes a teacher is constantly $0.6$. Both are unlikely to be true.
The probability of Max doing their homework on time is $0.6$. - Find the probability that Max does three or more of their next seven homeworks on time.
- 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?
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 $2$.
$0.415$
$X\sim B(15,p)$ - Given that $p = 0.4$, find $\mathrm{P}(X > 11)$
- Given instead that $\mathrm{P}(X = 0) = 0.03$, find $p$ correct to three decimal places.
$X\sim B(n,p)$. Given that the mean and standard deviation of $X$ are both equal to $0.9$, find $n$ and $p$.
$p = 0.1$ and $n = 9$
$X\sim B(n,0.2)$, and the mean is three times as large as the standard deviation. Find $n$.
$36$
The chance of a sweet being lemon flavoured is $10$%. Erin buys $20$ sweets. - Write down two conditions, in context, which must apply for a Binomial model to be valid for this situation.
- What is the probability that there is at least $1$ lemon flavoured sweet?
- 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.
- The chance of each sweet being lemon flavoured is independent of the next sweet; the probability of a sweet being lemon flavoured is 10% for each sweet.
- $0.878$
- $29$
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. - Find the probability that the coach will be full if the school invites $30$ students onto the trip.
- 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.
The probability of Jamie getting a detention on any given week is $0.1$. - In a term of $14$ weeks, find the probability Jamie receives more than $3$ detentions.
- After how many weeks will the average number of detentions Jamie has received exceed $5$?
- 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.
- $0.0441$
- $51$ weeks
- $0.259$
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. - Find the probability of an unbiased coin landing on heads $510$ times out of $1000$.
- Explain whether you agree with Dylan's claim.
- $0.02$
- Even though it is unlikely to get exactly $510$ heads, it is still one of the most likely outcomes and does not justify the claim.
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. - Find the distribution of $X$, the number of apple flavoured sweets selected.
- 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 $Y$.
- Probabilities: $\dfrac{8}{40},\dfrac{17}{40},\dfrac{15}{40}$ for $x = 0, 1, 2$
- Probabilities: $0.104, 0.580, 0.316$ for $y = 0, 1, 2$