Two independent categorical choices

1. In a sixth form college each student in Year 12 and Year 13 is either left-handed (L) or right-handed (R).

The partially completed tree diagram, where \(p\) is a probability, gives information about these students.
\includegraphics[max width=\textwidth, alt={}, center]{86446ce3-496a-4f02-9566-9b207bac9efa-10_960_981_477_543}

1. Complete the tree diagram, in terms of \(p\) where necessary. The probability that a student is left-handed is 0.11
2. Find the value of \(p\)
3. Find the probability that a student selected at random is in Year 12 and left-handed. Given that a student is right-handed,
4. find the probability that the student is in Year 12

1. In a school canteen, students can choose from a main course of meat ( \(M\) ), fish ( \(F\) ) or vegetarian ( \(V\) ). They can then choose a drink of either water ( \(W\) ) or juice ( \(J\) ).

The partially completed tree diagram, where \(p\) and \(q\) are probabilities, shows the probabilities of these choices for a randomly selected student. \section*{Drink} \begin{figure}[h] \end{figure}

1. Complete the tree diagram, giving your answers in terms of \(p\) and \(q\) where appropriate.
2. Find an expression, in terms of \(p\) and \(q\), for the probability that a randomly selected student chooses water to drink. The events "choosing a vegetarian main course" and "choosing water to drink" are independent.
3. Find a linear equation in terms of \(p\) and \(q\). A student who has chosen juice to drink is selected at random. The probability that they chose fish for their main course is \(\frac { 7 } { 30 }\)
4. Find the value of \(p\) and the value of \(q\). The canteen manager claims that students who choose water to drink are most likely to choose a fish main course.
5. State, showing your working clearly, whether or not the manager's claim is correct.

2. An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability \(\frac { 2 } { 3 }\) of landing heads is spun.
When a blue ball is selected a fair coin is spun.

1. Complete the tree diagram below to show the possible outcomes and associated probabilities.
   \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_785_385_744_568}
   \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_1054_483_760_954} Shivani selects a ball and spins the appropriate coin.
2. Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
3. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
4. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.

2. An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability \(\frac { 2 } { 3 }\) of landing heads is spun.
When a blue ball is selected a fair coin is spun.

1. Complete the tree diagram below to show the possible outcomes and associated probabilities.
   \includegraphics[max width=\textwidth, alt={}, center]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_787_395_734_548} \section*{Coin}
   \includegraphics[max width=\textwidth, alt={}]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_1007_488_808_950}
   Shivani selects a ball and spins the appropriate coin.
2. Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
3. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
4. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.

4

1. Chris shops at his local store on his way to and from work every Friday.
   The event that he buys a morning newspaper is denoted by \(M\), and the event that he buys an evening newspaper is denoted by \(E\). On any one Friday, Chris may buy neither, exactly one or both of these newspapers.
   1. Complete the table of probabilities, printed on the opposite page, where \(M ^ { \prime }\) and \(E ^ { \prime }\) denote the events 'not \(M\) ' and 'not \(E\) ' respectively.
   2. Hence, or otherwise, find the probability that, on any given Friday, Chris buys exactly one newspaper.
   3. Give a numerical justification for the following statement.
      'The events \(M\) and \(E\) are not mutually exclusive.'
2. The event that Chris buys a morning newspaper on Saturday is denoted by \(S\), and the event that he buys a morning newspaper on the following day, Sunday, is denoted by \(T\). The event that he buys a morning newspaper on both Saturday and Sunday is denoted by \(S \cap T\). Each combination of the events \(S\) and \(T\) is independent of any combination of the events \(M\) and \(E\). However, the events \(S\) and \(T\) are not independent, with $$\mathrm { P } ( S ) = 0.85 , \quad \mathrm { P } ( T \mid S ) = 0.20 \quad \text { and } \quad \mathrm { P } \left( T \mid S ^ { \prime } \right) = 0.75$$ Find the probability that, on a particular Friday, Saturday and Sunday, Chris buys:
   1. all four newspapers;
   2. none of the four newspapers.
3. 1. State, as briefly as possible, in the context of the question, the event that is denoted by \(M \cap E ^ { \prime } \cap S \cap T ^ { \prime }\).
   2. Calculate the value of \(\mathrm { P } \left( M \cap E ^ { \prime } \cap S \cap T ^ { \prime } \right)\). \section*{Answer space for question 4}

4. 1. \cline { 2 - 4 } \multicolumn{1}{c|}{}	\(\boldsymbol { M }\)	\(\boldsymbol { M } ^ { \prime }\)	Total
      \(\boldsymbol { E }\)	0.16		0.28
      \(\boldsymbol { E } ^ { \prime }\)
      Total		0.60	1.00

      \includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-11_2050_1707_687_153}