2.03a Mutually exclusive and independent events

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]

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 discrete random variable \(X\) takes values 1, 2, 3, 4 and 5, and its probability distribution is defined as follows. $$\mathrm{P}(X = x) = \begin{cases} a & x = 1, \\ \frac{1}{2}\mathrm{P}(X = x - 1) & x = 2, 3, 4, 5, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac{16}{31}\). [2]

The discrete probability distribution for \(X\) is given in the table.

1. Find the probability that \(X\) is odd. [1]

Two independent values of \(X\) are chosen, and their sum \(S\) is found.

1. Find the probability that \(S\) is odd. [2]
2. Find the probability that \(S\) is greater than 8, given that \(S\) is odd. [3]

Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows. $$\mathrm{P}(Y = y + 1) = \frac{1}{2}\mathrm{P}(Y = y) \quad \text{for all positive integers } y.$$

1. Find P\((Y = 1)\). [2]
2. Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car. [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]

In a large examination room each candidate has just one electronic calculator.

• \(G\) is the event that a randomly chosen candidate has a graphical calculator.
• \(T\) is the event that a randomly chosen candidate has a 'Texio' brand calculator.

You are given the following probabilities. $$\text{P}(G) = 0.65 \quad \text{P}(T) = 0.4 \quad \text{P}(G \cap T) = 0.25$$

1. Are the events \(G\) and \(T\) independent? Justify your answer with an appropriate calculation. [2]
2. Find P(\(T | G\)) and explain, in the context of this question, what this probability represents. [3]

\(A\) 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]

\(A\) and \(B\) are independent events. \(P(A) = \frac{3}{4}\) and \(P(A \cap B) = \frac{1}{4}\). Find \(P(A' \cap B)\) and \(P(A' \cap B')\). [5]