Calculate the expected number of people who will pass this polygraph test.

Determine the probability that fewer than $7$ people will pass this polygraph test.

only a sandwich.

Copy and complete the probability distribution table for Y.

an estimate for the standard deviation of the number of emails received per working day.

Find the expected number of bags in this crate that contain at most one small apple.

Find the probability that an orange weighs between 289 g and 310 g.

Suppose that the probability of Archie receiving more than 10 emails in total on any one day is 0.99. Find the value of λ.

Six coins are tossed simultaneously 320 times, with the following results.

At the 5% level of significance, test the hypothesis that all the coins are fair.

State an assumption that is being made for $X$ to be considered as following a binomial distribution.

Find the maximum number of tickets that could be sold if the expected number of passengers who arrive to board the flight must be less than or equal to $72$.

Write down the expected number of passengers who will arrive to board the flight if $72$ tickets are sold.

Calculate the probability that exactly $4$ people will fail this polygraph test.

median reaction time.

Show that Jonas’s network satisfies the requirement of there being less than a $2\%$ probability of the network failing after a power surge.

Find the probability that Arianne throws two consecutive darts that land more than $15 \text{cm}$ from $\text{O}$.

The airline sells $74$ tickets for this flight. Find the probability that more than $72$ passengers arrive to board the flight.

Find the probability that Arianne throws two consecutive darts that land more than $15 \text{cm}$ from $\text{O}$.

Find the probability he plays between $13 \text{minutes}$ and $18 \text{minutes}$ in one game and more than $20 \text{minutes}$ in the other game.

Using this distribution model, find the standard deviation of $X$.

an estimate for the mean number of emails received per working day.

A customer is selected at random. Find the probability that the customer is male and buys a sandwich.

Calculate the probability that, on a randomly selected day, Timmy makes a profit.

Write down the probability of drawing three blue marbles.

A randomly selected participant has a reaction time greater than 0.65 seconds. Find the probability that the participant is in Group X.

Find the probability that at least 48 bags in this crate contain at most one small apple.

It is further given that P(X ≤ 1) = 0.09478 correct to 4 significant figures.

Determine the value of n and the value of p.

both a sandwich and a cake.

Find the probability that more than 100 cakes will be sold on a typical day.

Find the remainder when $p\left(x\right)$ is divided by $\left(x-2\right)$.

Write down the value of $x$.

Hence, find the value of $y$.

Write down the number of low production hives.

Using this distribution model, find .

Find $\text{Var}(X)$.

Find $\text{P}(Y⩾3)$.

To the nearest gram, find the minimum weight of an orange that the grocer will buy.

Find the expected number of cakes sold on a typical day.

Hence, find the probability that fewer than $k$ students are left handed.

Calculate an appropriate value of ${\chi }^2$ and state your conclusion, using a 1% significance level.

State suitable hypotheses for testing this belief.

Find $\text{E}(Y)$.

Now suppose that Archie received exactly 20 emails in total in a consecutive two day period. Show that the probability that he received exactly 10 of them on the first day is independent of λ.

Find the probability of rolling two or more red faces.

Find the probability of rolling exactly one red face.

During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.

Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.

Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75 that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regular production hives.

Give one piece of evidence that suggests Willow’s Poisson distribution model is not a good fit.

Given that $\text{P}(X=2)=0.241667$ and $\text{P}(X=3)=0.112777$, use part (a) to find the value of $\mu$.

Write down the value of $a$ and of $b$.

Hence, find the value of σ.

The shop is open for 24 days every month.

Calculate the probability that, in a randomly selected month, Timmy makes a profit on between 5 and 10 days (inclusive).

Find the probability that the grocer buys more than half the oranges in a box selected at random.

Hence, find the probability that exactly $k$ students are left handed;

Find the value of $k$;

Find $\text{E}(X)$.

Find the most likely number of taxis that would arrive during T.

Find the standardized value for 289 g.

Use this regression line to estimate the monthly honey production from a hive that has 270 bees.

Hence estimate $p$, the probability that a randomly chosen egg is brown.

Find the probability that exactly 4 taxis arrive during T.

Ten of the participants with reaction times greater than 0.65 are selected at random. Find the probability that at least two of them are in Group X.

By calculating an appropriate ${\chi }^2$ statistic, test, at the 5% significance level, whether or not the binomial distribution gives a good fit to these data.

Find the probability that on a randomly selected day, Steffi does not visit Will’s house.

Find the number of hives that have a high production.

Find the least possible value of n.

Find the remainder when $p\left(x\right)$ is divided by $\left(x-3\right)$.

Find the probability that on any given day Mr Burke chooses a female student to answer a question.

Write down the probability that the first disc selected is red.

Write down the value of $n$ and the value of $p$.

The mean number of squirrels in a certain area is known to be 3.2 squirrels per hectare of woodland. Within this area, there is a 56 hectare woodland nature reserve. It is known that there are currently at least 168 squirrels in this reserve.

Assuming the population of squirrels follow a Poisson distribution, calculate the probability that there are more than 190 squirrels in the reserve.

Find the expected number of weeks in the year in which Lucca eats no bananas.

Find $k$.

Sketch a diagram to represent this information.

Show that $m=15.7$.

Find the probability that Emlyn plays between $13 \text{minutes}$ and $18 \text{minutes}$ in a game.

Find the probability that Emlyn plays more than $20 \text{minutes}$ in a game.

Find the value of $x$.

Find the expected number of successful shots Emlyn will make on Monday, based on the results from Saturday and Sunday.

Emlyn claims the results from Saturday and Sunday show that his expected number of successful shots will be more than Johan’s.

Determine if Emlyn’s claim is correct. Justify your reasoning.

Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.

Prove that $p\left(x\right)$ has only one real zero.

Find the significance level of this procedure.

Show that $\text{P}(X=x+1)=\frac{\mu }{x+1}\times \text{P}(X=x),x\in \mathbb{N}$.

Write down the transformation that will transform the graph of $y=p\left(x\right)$ onto the graph of $y=8x^3-12x^2+16x-24$.

Determine the mean of X.

Show that μ = 106.

Show that the probability that Chloe wins the game is $\frac{3}{8}$.

Show that, after a turn, the probability that Ted adds exactly $10 to his winnings is $\frac{1}{3}$.

The Head of Year, Mrs Smith, decides to select a student at random from the year group to read the notices in assembly. There are 80 students in total in the year group. Mrs Smith calculates the probability of picking a male student 8 times in the first 20 assemblies is 0.153357 correct to 6 decimal places.

Find the number of male students in the year group.

Some time later, the actual value of $\mu$ is 503. Find the probability of a Type II error.

Find the probability he will choose a female student 8 times.

Find $q$.

Find the probability he will choose a male student at most 9 times.

Determine the variance of X.

In any given year of 365 days, the probability that Steffi does not visit Will for at most $n$ days in total is 0.5 (to one decimal place). Find the value of $n$.

A female customer is selected at random. Find the probability that she buys a sandwich.

Find P(M < 95) .

Jill plays the game nine times. Find the probability that she wins exactly two prizes.

Explain why the probability of drawing three white marbles is $\frac{1}{6}$.

The random variable $X$ follows a Poisson distribution with a mean of $\lambda$ and $6\text{P}\left(X=3\right)=3\text{P}\left(X=2\right)-2\text{P}\left(X=1\right)+3\text{P}\left(X=0\right)$.

Find the value of $\lambda$.

The grocer selects two boxes at random.

Find the probability that the grocer buys more than half the oranges in each box.

Find the probability that a bag of apples selected at random contains at most one small apple.

Calculate the mean of these data and hence estimate the value of $p$.

Find $p$.

Write down the value of k.

Show that $a=32$ and $b=\frac{1}{12}$.

The firm obtains a significant result when comparing section $2$ of the written assessment and attribute $\text{X}$. Interpret this result.

Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.

Let $X$ be the number of red discs selected. Find the smallest value of $m$ for which .

Find the probability that Lucca eats at least one banana in a particular day.

Hence find the expected number of times per day that Steffi is fed at Will’s house.

The bag contains a total of ten marbles of which $w$ are white. Find $w$.

Find the median of $X$.

Ten of the cats are chosen at random. Find the probability that exactly one of them weighs over $6.25 \text{kg}$.

Find the probability he will choose a female student 8 times.

Calculate the mean number of brown eggs in a box.

Let $N$ be the number of cats weighing over $4.62 \text{kg}$.

Find the variance of $N$.

Ted will always have another turn if he expects an increase to his winnings.

Find the least value of $w$ for which Ted should end the game instead of having another turn.

The tests are performed at the $5\%$ significance level.

Assuming that:

• there is no correlation between the marks in any of the sections and scores in any of the attributes,
• the outcome of each hypothesis test is independent of the outcome of the other hypothesis tests,

find the probability that at least one of the tests will be significant.

Find the probability that exactly one of them weighs over $4.62 \text{kg}$.

Find the probability that Arianne scores at least $5$ points in the competition.

Find the probability of scoring more than $2$ sixes when this die is rolled $5$ times.

Given that Arianne scores at least $5$ points, find the probability that Arianne scores less than $8$ points.

a dart lands less than $13 \text{cm}$ from $\text{O}$.

a dart lands more than $15 \text{cm}$ from $\text{O}$.

Find the probability that Arianne does not score a point on a turn of three darts.

Find the probability that Arianne scores at least $5$ points and less than $8$ points.

Given that Arianne scores at least $5$ points, find the probability that Arianne scores less than $8$ points.

a dart lands less than $13 \text{cm}$ from $\text{O}$.

a dart lands more than $15 \text{cm}$ from $\text{O}$.

Find the probability that Arianne does not score a point on a turn of three darts.

Find Arianne’s expected score in the competition.

Find the probability that Arianne scores at least $5$ points in the competition.

Find the probability that Arianne scores at least $5$ points and less than $8$ points.

Write down null and alternative hypotheses for Loreto’s test.

Using the data from Loreto’s sample, perform the hypothesis test at a $5\%$ significance level to determine if Loreto should employ extra staff.

Write down null and alternative hypotheses for this test.

Perform the test, clearly stating the conclusion in context.