2.03c Conditional probability: using diagrams/tables

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]

Val buys electrical components from one of 3 suppliers \(A\), \(B\), \(C\), in the ratio \(2:1:7\). The probability that the component is faulty is \(0.33\) for \(A\), \(0.45\) for \(B\) and \(0.05\) for \(C\). Val selects a component at random.

1. Find the probability that the component works. [3]
2. Given that the component works, find the probability that Val bought the component from supplier \(B\). [2]

Events \(A\) and \(B\) are such that \(P(A \cup B) = 0.95\), \(P(A \cap B) = 0.6\) and \(P(A|B) = 0.75\).

1. Find \(P(B)\). [3]
2. Find \(P(A)\). [3]
3. Show that the events \(A'\) and \(B\) are independent. [2]

Only two airlines fly daily into an airport. AMP Air has 70 flights per day and Volt Air has 65 flights per day. Passengers flying with AMP Air have an 18% probability of losing their luggage and passengers flying with Volt Air have a 23% probability of losing their luggage. You overhear a passenger in the airport complaining about her luggage being lost. Find the exact probability that she travelled with Volt Air, giving your answer as a rational number. [6]

\(P(E) = 0.25\), \(P(F) = 0.4\) 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]

Miguel has six numbered tiles, labelled 2, 2, 3, 3, 4, 4. He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.

1. Show that \(P(M = 6) = \frac{1}{3}\) [2]

The table shows the probability distribution of \(M\)

\(m\)	4	5	6	7	8
\(P(M = m)\)	\(\frac{1}{15}\)	\(\frac{4}{15}\)	\(\frac{1}{3}\)	\(\frac{4}{15}\)	\(\frac{1}{15}\)

Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.

1. Find \(P(M = S)\) [3]
2. Find \(P(S = 7 | M = S)\) [4]

A manufacturing plant produces electronic circuit boards that need to pass two quality checks - a mechanical inspection and an electrical test. Historical data shows that 15% of boards fail the mechanical inspection. Of those that pass the mechanical inspection, 8% fail the electrical test. Of those that fail the mechanical inspection, 60% fail the electrical test.

1. If a board is randomly selected from production, what is the probability that it passes both inspections? [2]
2. If a board is selected at random and is found to have passed the electrical test, what is the probability that it also passed the mechanical inspection? [3]
3. The company continues to test boards from a large batch until finding one that passes both inspections. Each board is tested independently of all others. What is the probability that they need to test exactly 3 boards to find one that passes both inspections? [3]

Miguel has six numbered tiles, labelled 2, 2, 3, 3, 4, 4. He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.

1. Show that \(P(M = 6) = \frac{1}{3}\) [2]

The table shows the probability distribution of \(M\) Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.

1. Find \(P(M = S)\) [3]
2. Find \(P(S = 7 | M = S)\) [4]

A factory buys 10\% of its components from supplier \(A\), 30\% from supplier \(B\) and the rest from supplier \(C\). It is known that 6\% of the components it buys are faulty. Of the components bought from supplier \(A\), 9\% are faulty and of the components bought from supplier \(B\), 3\% are faulty.

1. Find the percentage of components bought from supplier \(C\) that are faulty. [3]

A component is selected at random.

1. Explain why the event "the component was bought from supplier \(B\)" is not statistically independent from the event "the component is faulty". [1]

The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students. \includegraphics{figure_7} A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and the second student studies Geography but not Psychology. [4]

\(A\) and \(B\) are two events. You are given that \(\mathrm{P}(A) = 0.6\), \(\mathrm{P}(B) = 0.5\) and \(\mathrm{P}(A \cup B) = 0.8\).

1. Find \(\mathrm{P}(A \cap B)\). [2]
2. Find \(\mathrm{P}(B | A)\). [2]
3. Explain whether the events \(A\) and \(B\) are independent or not. [1]

The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{4}{\pi(1+x^2)} & 0 \leq x \leq 1, \\ 0 & \text{otherwise.} \end{cases}$$

1. Verify that the median value of \(X\) lies between 0.41 and 0.42. [3]
2. Show that E\((X) = \frac{2}{\pi}\ln 2\). [2]
3. Find Var\((X)\). [5]
4. Given that \(\tan\frac{1}{8}\pi = \sqrt{2} - 1\), find the exact value of P(\(X > \frac{1}{4}\sqrt{3}|X > \sqrt{2} - 1\)). [3]