The numbers of books read in the past $3$ months by the members of a reading club are shown in the following table.

No. books	$<2$	$2-4$	$5-7$	$8-9$	$10$	$>10$
No. members $\left(f\right)$	$3$	$5$	$22$	$7$	$2$	$1$

Find the probability that three randomly selected members have all read fewer than eight books, given that they have all read more than four books.

One hundred people are attending a conference. The following Venn diagram shows how many are male $\left(M\right)$, have brown eyes $\left(BE\right)$ and are right-handed $\left(RH\right)$.

Given that there are $43$ males with brown eyes, $42$ right-handed males and 46 right-handed people with brown eyes, copy and complete the Venn diagram.

One hundred people are attending a conference. The following Venn diagram shows how many are male $\left(M\right)$, have brown eyes $\left(BE\right)$ and are right-handed $\left(RH\right)$.

Two attendees are selected at random. Find the probability that they are both females who are not right-handed.       [correct up to five decimal places]

One hundred people are attending a conference. The following Venn diagram shows how many are male $\left(M\right)$, have brown eyes $\left(BE\right)$ and are right-handed $\left(RH\right)$.

Two attendees are selected at random. Find the probability that exactly one of them is right-handed, given that neither of them have brown eyes.      [correct up to three decimal places]

A student travels to college by either of two routes, $A$ or $B$. The probability that they use route $A$ is $0.3$, and the probability that they are passed by a bus on their way to college on any particular day is $0.034$. They are twice as likely to be passed by a bus when they use route $B$ as when they use route $A$.

Use the tree diagram opposite to form and solve a pair of simultaneous equations in $x$ and $y$.

A student travels to college by either of two routes, $A$ or $B$. The probability that they use route $A$ is $0.3$, and the probability that they are passed by a bus on their way to college on any particular day is $0.034$. They are twice as likely to be passed by a bus when they use route $B$ as when they use route $A$.

Find the probability that the student uses route $B$, given that they are not passed by a bus on their way to college.   [correct up to three decimal places]

Two ordinary fair dice are rolled. If the first shows a number less than $3$ , then the score is the mean of the numbers obtained; otherwise the score is equal to half the absolute (non-negative) difference between the numbers obtained. Find the probability that the score is:

positive

Two ordinary fair dice are rolled. If the first shows a number less than $3$ , then the score is the mean of the numbers obtained; otherwise the score is equal to half the absolute (non-negative) difference between the numbers obtained. Find the probability that the score is:

greater than $1$ , given that it is less than $2$