Venn diagram with independence constraint

3 Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.

• \(A\) is the event that Isobel's parents watch a match.
• \(\quad B\) is the event that Isobel scores in a match.

You are given that \(\frac { 3 } { 7 }\) and \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).

1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
4. By comparing \(\mathrm { P } ( B \mid A )\) with \(\mathrm { P } ( B )\), explain why Isobel should ask her parents not to attend.

2. In the Venn diagram below, \(A , B\) and \(C\) are events and \(p , q , r\) and \(s\) are probabilities. The events \(A\) and \(C\) are independent and \(\mathrm { P } ( A ) = 0.65\) \includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-04_373_815_397_568}

1. State which two of the events \(A\), \(B\) and \(C\) are mutually exclusive.
2. Find the value of \(r\) and the value of \(s\). The events ( \(A \cap C ^ { \prime }\) ) and ( \(B \cup C\) ) are also independent.
3. Find the exact value of \(p\) and the exact value of \(q\). Give your answers as fractions.

4. The Venn diagram shows the probabilities of customer bookings at Harry's hotel. \(R\) is the event that a customer books a room \(B\) is the event that a customer books breakfast \(D\) is the event that a customer books dinner \(u\) and \(t\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{e3b92a5b-c0ad-4176-9b05-cb07a44aa265-08_604_1047_696_450}

1. Write down the probability that a customer books breakfast but does not book a room. Given that the events \(B\) and \(D\) are independent
2. find the value of \(t\)
3. hence find the value of \(u\)
4. Find
   1. \(\quad\) P( \(D \mid R \cap B\) )
   2. \(\mathrm { P } \left( D \mid R \cap B ^ { \prime } \right)\) A coach load of 77 customers arrive at Harry's hotel. Of these 77 customers 40 have booked a room and breakfast 37 have booked a room without breakfast
5. Estimate how many of these 77 customers will book dinner.

The following Venn diagram shows the participation of 100 students in three activities, \(A\), \(B\), and \(C\), which represent athletics, baseball and climbing respectively. \includegraphics{figure_3} For these 100 students, participation in athletics and participation in climbing are independent events.

1. Show that \(x = 10\) and find the value of \(y\). [5]
2. Two students are selected at random, one after the other without replacement. Find the probability that the first student does athletics and the second student does only climbing. [3]