Determine the value of $b$.

Find the value of $a$, providing evidence for your answer.

Show that $2k^2-k+0.12=0$.

Write down the probability of drawing three blue marbles.

Calculate the expected value of the score.

The die is rolled 80 times. On how many rolls would you expect to obtain a three?

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

Copy and complete the probability distribution table for Y.

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

Write down the value of $x$.

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

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

Find the probability, in terms of $n$, that the game will end on her second draw.

Find the value of p.

Find the value of $k$.

Hence, find the value of $y$.

fourth draw.

Sketch the graph of $f$. State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of $a$.

the median of $X$.

Given that $\tan \theta >0$, find $\tan \theta$.

Find the expected value of T.

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

Show that μ = 106.

Write down $\text{P}(X=2)$.

third draw.

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Write down the value of k.

Find $q$.

Find the difference between the greatest possible expected value and the least possible expected value.

Find the values of the constants $a$ and $b$.

Let $y=\frac{1}{\cos x}$, for . The graph of $y$between $x=\theta$ and $x=\frac{\pi }{4}$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Determine the value of $\text{E}(X^2)$.

Find the probability, in terms of $n$, that the game will end on her first draw.

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 $n$.

$\text{E}\left(X\right)$.

The following table shows the probability distribution of a discrete random variable $X$ where $x=1, 2, 3, 4$.

Find the value of $k$, justifying your answer.

Find $\text{P}(X>2)$.

Determine the minimum value of $x$ such that the probability Kati receives at least one free gift is greater than 0.5.

Find the value of a and the value of b.

Find, in terms of $a$, the probability that $X$ lies between 1 and 3.

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

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

Find the value of $p$.

Hence, find the value of $\text{E}\left(X\right)$.

State the range of possible values of $r$.

Hence, find the range of possible values for $\text{E}\left(Y\right)$.

Find the value of $c$.

Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of $n$ for the value of the grand prize in the $n\text{th}$ week of the lottery.

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

Find μ, the expected value of X.

Find P(X > μ).

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

Show that $\text{P}(X=3)=0.001$ and $\text{P}(X=4)=0.0027$.

Find the probability of rolling two or more red faces.

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

Deduce that $\frac{\text{P}(X=n)}{\text{P}(X=n-1)}=\frac{0.9(n-1)}{n-3}$ for $n>3$.

Complete the probability distribution table for $X$.

Find the value of $\text{Var}(X)$.

Find the value of $k$.

Hayley plays the game when $n$ = 5. She pays $20 to play and can earn money back depending on the number of draws it takes to obtain a blue marble. She earns no money back if she obtains a blue marble on her first draw. Let M be the amount of money that she earns back playing the game. This information is shown in the following table.

Find the value of $k$ so that this is a fair game.

$a$.

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

The $w\text{th}$ week is the first week in which the player is expected to make a profit. Ryan knows that if he buys a lottery ticket in the $w\text{th}$ week, his expected profit is $\$p$.

Find the value of $p$.

Hence, find the range of possible values of $q$.

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

Given that $\text{E}\left(X\right)=10$, find the value of $N$.

A discrete random variable $X$ has the probability distribution given by the following table.

Given that $\text{E}\left(X\right)=\frac{19}{12}$, determine the value of $p$ and the value of $q$.

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Find $\text{P}\left(\mu -1.5\sigma <X<\mu +1.5\sigma \right)$.

Find the value of $\mu$ and of $\sigma$.

The selling prices of the different categories of avocado at this supermarket are shown in the following table:

The supermarket pays the farm $\$ 200$ for the avocados and assumes it will then sell them in exactly the same proportion as purchased from the farm.

According to this model, find the minimum number of avocados that must be sold so that the net profit for the supermarket is at least $\$ 438$.

Show that $p=0.4$ and $q=0.2$.

Find $p$.

Show that $b=0.3-a$.

Find the probability of rolling exactly one red face.

Find the expected value of $X$.

Show that $\cos \theta =\frac{3}{4}$.

Find the value of $p$ which gives the largest value of $\text{E}\left(X\right)$.

Hence, find the largest value of $\text{E}\left(X\right)$.

Find the value of k.

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.

Find the value of $p$.

Find an expression for $q$ in terms of $p$.

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

(i)     Hence show that $X$ has two modes $m_1$ and $m_2$.

(ii)     State the values of $m_1$ and $m_2$.

Find $\text{P}(X=2|X>0)$.

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

Agnes and Barbara play a game using these dice. Agnes rolls die $\text{A}$ once and Barbara rolls die $\text{B}$ once. The probability that Agnes’ score is less than Barbara’s score is $\frac{1}{2}$.

Find the value of $\text{E}\left(Y\right)$.

Determine whether this lottery is a fair game in the first week. Justify your answer.

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