2.03b Probability diagrams: tree, Venn, sample space

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]

\(P(E) = 0.25\), \(P(F) = 0.4\) and \(P(E \cap F) = 0.12\)

1. Find \(P(E'|F')\) [2 marks]
2. Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer. [2 marks]

The event \(G\) has \(P(G) = 0.15\) The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.

1. Draw a Venn diagram to illustrate the events \(E\), \(F\) and \(G\), giving the probabilities for each region. [3 marks]
2. Find \(P([F \cup G]')\) [2 marks]

A group of students were surveyed by a principal and \(\frac{2}{3}\) were found to always hand in assignments on time. When questioned about their assignments \(\frac{3}{5}\) said they always start their assignments on the day they are issued and, of those who always start their assignments on the day they are issued, \(\frac{11}{20}\) hand them in on time.

1. Draw a tree diagram to represent this information. [3 marks]
2. Find the probability that a randomly selected student:
   1. always start their assignments on the day they are issued and hand them in on time. [2 marks]
   2. does not always hand in assignments on time and does not start their assignments on the day they are issued. [4 marks]
3. Determine whether or not always starting assignments on the day they are issued and handing them in on time are statistically independent. Give reasons for your answer. [2 marks]

Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table. Four puppies are chosen at random to train as guide dogs.

1. Determine the probability that at least 3 black puppies are chosen. [3]
2. Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen. [3]
3. Explain whether the 2 events 'choosing exactly 3 females' and 'choosing at least 3 black puppies' are independent events. [1]

Mr Jones has 3 tins of beans and 2 tins of pears. His daughter has removed the labels for a school project, and the tins are identical in appearance. Mr Jones opens tins in turn until he has opened at least 1 tin of beans and at least 1 tin of pears. He does not open any remaining tins.

1. Draw a tree diagram to illustrate this situation, labelling each branch with its associated probability. [3]
2. Find the probability that Mr Jones opens exactly 3 tins. [3]
3. It is given that the last tin Mr Jones opens is a tin of pears. Find the probability that he opens exactly 3 tins. [5]

Each of the 30 students in a class plays at least one of squash, hockey and tennis. • 18 students play squash • 19 students play hockey • 17 students play tennis • 8 students play squash and hockey • 9 students play hockey and tennis • 11 students play squash and tennis

1. Find the number of students who play all three sports. [3]

A student is picked at random from the class.

1. Given that this student plays squash, find the probability that this student does not play hockey. [1]

Two different students are picked at random from the class, one after the other, without replacement.

1. Given that the first student plays squash, find the probability that the second student plays hockey. [4]