2.03a Mutually exclusive and independent events

A bag contains a large number of coloured counters. Each counter is labelled A, B or C 30% of the counters are labelled A 45% of the counters are labelled B The rest of the counters are labelled C It is known that 2% of the counters labelled A are red 4% of the counters labelled B are red 6% of the counters labelled C are red One counter is selected at random from the bag.

1. Complete the tree diagram on the opposite page to illustrate this information. [2]
2. Calculate the probability that the counter is labelled A and is not red. [2]
3. Calculate the probability that the counter is red. [2]
4. Given that the counter is red, find the probability that it is labelled C [3]

\includegraphics{figure_3}

Three events \(A\), \(B\) and \(C\) are such that $$\mathrm{P}(A) = 0.1 \quad \mathrm{P}(B|A) = 0.3 \quad \mathrm{P}(A \cup B) = 0.25 \quad \mathrm{P}(C) = 0.5$$ Given that \(A\) and \(C\) are mutually exclusive

1. find P\((A \cup C)\) [1]
2. Show that P\((B) = 0.18\) [3]

Given also that \(B\) and \(C\) are independent,

1. draw a Venn diagram to represent the events \(A\), \(B\) and \(C\) and the probabilities associated with each region. [5]

A contractor bids for two building projects. He estimates that the probability of winning the first project is 0.5, the probability of winning the second is 0.3 and the probability of winning both projects is 0.2.

1. Find the probability that he does not win either project. [3]
2. Find the probability that he wins exactly one project. [2]
3. Given that he does not win the first project, find the probability that he wins the second. [2]
4. By calculation, determine whether or not winning the first contract and winning the second contract are independent events. [3]

Jake and Kamil are sometimes late for school. The events \(J\) and \(K\) are defined as follows \(J =\) the event that Jake is late for school \(K =\) the event that Kamil is late for school \(\text{P}(J) = 0.25\), \(\text{P}(J \cap K) = 0.15\) and \(\text{P}(J' \cap K') = 0.7\) On a randomly selected day, find the probability that

1. at least one of Jake or Kamil are late for school, [1]
2. Kamil is late for school. [2]

Given that Jake is late for school,

1. find the probability that Kamil is late. [3]

The teacher suspects that Jake being late for school and Kamil being late for school are linked in some way.

1. Determine whether or not \(J\) and \(K\) are statistically independent. [2]
2. Comment on the teacher's suspicion in the light of your calculation in (d). [1]

There are 125 sixth-form students in a college, of whom 60 are studying only arts subjects, 40 only science subjects and the rest a mixture of both. Three students are selected at random, without replacement. Find the probability that

1. all three students are studying only arts subjects, [4]
2. exactly one of the three students is studying only science subjects. [3]

The events \(A\) and \(B\) are independent such that \(P(A) = 0.25\) and \(P(B) = 0.30\). Find

1. \(P(A \cap B)\), [2]
2. \(P(A \cup B)\), [2]
3. \(P(A | B')\). [4]

For any married couple who are members of a tennis club, the probability that the husband has a degree is \(\frac{3}{5}\) and the probability that the wife has a degree is \(\frac{1}{2}\). The probability that the husband has a degree, given that the wife has a degree, is \(\frac{11}{12}\). A married couple is chosen at random.

1. Show that the probability that both of them have degrees is \(\frac{11}{24}\). [2]
2. Draw a Venn diagram to represent these data. [5]

Find the probability that

1. only one of them has a degree, [2]
2. neither of them has a degree. [3]

Two married couples are chosen at random.

1. Find the probability that only one of the two husbands and only one of the two wives have degrees. [6]

Sixteen cards have been lost from a pack, which therefore contains only 36 cards. Two cards are drawn at random from the pack. The probability that both cards are red is \(\frac{1}{3}\).

1. Show that \(r\), the number of red cards in the pack, satisfies the equation $$r(r - 1) = 420.$$ [4 marks]
2. Hence or otherwise find the value of \(r\). [3 marks]
3. Find the probability that, when three cards are drawn at random from the pack,
   1. at least two are red, [6 marks]
   2. the first one is red given that at least two are red. [4 marks]

The events \(A\) and \(B\) are such that P\((A \cap B) = 0.24\), P\((A \cup B) = 0.88\) and P\((B) = 0.52\).

1. Find P\((A)\). [3 marks]
2. Determine, with reasons, whether \(A\) and \(B\) are
   1. mutually exclusive,
   2. independent.
   [4 marks]
3. Find P\((B | A)\). [2 marks]
4. Find P\((A' | B')\). [3 marks]

\(A\), \(B\) and \(C\) are three events such that \(\text{P}(A) = x\), \(\text{P}(B) = y\) and \(\text{P}(C) = x + y\). It is known that \(\text{P}(A \cup B) = 0.6\) and \(\text{P}(B \mid A) = 0.2\).

1. Show that \(4x + 5y = 3\). [2 marks]

It is also known that \(B\) and \(C\) are mutually exclusive and that \(\text{P}(B \cup C) = 0.9\)

1. Obtain another equation in \(x\) and \(y\) and hence find the values of \(x\) and \(y\). [4 marks]
2. Deduce whether or not \(A\) and \(B\) are independent events. [2 marks]

A darts player throws two darts, attempting to score a bull's-eye with each. The probability that he will achieve this with his first dart is \(0.25\). If he misses with his first dart, the probability that he will also miss with his second dart is \(0.7\). The probability that he will miss with at least one dart is \(0.9\).

1. Show that the probability that he succeeds with his first dart but misses with his second is \(0.15\). [5 marks]
2. Find the conditional probability that he misses with both darts, given that he misses with at least one. [3 marks]

Given that \(P(A \cup B) = 0.65\), \(P(A \cap B) = 0.15\) and \(P(A) = 0.3\), determine, with explanation, whether or not the events \(A\) and \(B\) are

1. mutually exclusive, [1 mark]
2. independent. [3 marks]

The students in a large Sixth Form can choose to do exactly one of Community Service, Games or Private Study on Wednesday afternoons. The probabilities that a randomly chosen student does Games and Private Study are \(\frac{3}{8}\) and \(\frac{1}{5}\) respectively. It may be assumed that the number of students is large enough for these probabilities to be treated as constant.

1. Find the probability that a randomly chosen student does Community Service. [2 marks]
2. If two students are chosen at random, find the probability that they both do the same activity. [3 marks]
3. If three students are chosen at random, find the probability that exactly one of them does Games. [3 marks]

Two-fifths of the students are girls, and a quarter of these girls do Private Study.

1. Find the probability that a randomly chosen student who does Private Study is a boy. [5 marks]

The diagram shows five cards, each with a letter on it. \includegraphics{figure_6} The letters A and E are vowels; the letters B, C and D are consonants.

1. Two of the five cards are chosen at random, without replacement. Find the probability that they both have vowels on them. [2]
2. The two cards are replaced. Now three of the five cards are chosen at random, without replacement. Find the probability that they include exactly one card with a vowel on it. [3]
3. The three cards are replaced. Now four of the five cards are chosen at random without replacement. Find the probability that they include the card with the letter B on it. [2]

In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random. • \(G\) is the event that this person goes to the gym. • \(R\) is the event that this person goes running. You are given that P(G) = 0.24, P(R) = 0.13 and P(G ∩ R) = 0.06.

1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(G\) and \(R\) are independent. [2]
3. Find P(R | G). [3]

The table shows all the possible products of the scores on two fair four-sided dice.

1. Find the probability that the product of the two scores is less than 10. [1]
2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]

In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that P(\(W\)) = 0.14, P(\(F\)) = 0.41 and P(\(W \cap F\)) = 0.11.

1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(W\) and \(F\) are independent. [2]
3. Find P(\(W\) | \(F\)) and explain what this probability represents. [3]

Each weekday, Marta travels to school by bus. Sometimes she arrives late. • \(L\) is the event that Marta arrives late. • \(R\) is the event that it is raining. You are given that \(\mathrm{P}(L) = 0.15\), \(\mathrm{P}(R) = 0.22\) and \(\mathrm{P}(L \mid R) = 0.45\).

1. Use this information to show that the events \(L\) and \(R\) are not independent. [1]
2. Find \(\mathrm{P}(L \cap R)\). [2]
3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram. [3]

The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{5}{16}, \text{P}(B) = \frac{1}{2} \text{ and P}(A|B) = \frac{1}{4}$$ Find

1. P\((A \cap B)\). [2]
2. P\((B'|A)\). [3]
3. P\((A' \cup B)\). [2]
4. Determine, with a reason, whether or not the events \(A\) and \(B\) are independent. [3]

The events \(A\) and \(B\) are independent and such that $$\text{P}(A) = 2\text{P}(B) \text{ and } \text{P}(A \cap B) = \frac{1}{8}.$$

1. Show that \(\text{P}(B) = \frac{1}{4}\). [5 marks]
2. Find \(\text{P}(A \cup B)\). [3 marks]
3. Find \(\text{P}(A | B')\). [2 marks]

In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random. • \(T\) is the event that this person likes tomato soup. • \(M\) is the event that this person likes mushroom soup. You are given that \(\text{P}(T) = 0.55\), \(\text{P}(M) = 0.33\) and \(\text{P}(T|M) = 0.80\).

1. Use this information to show that the events \(T\) and \(M\) are not independent. [1]
2. Find \(\text{P}(T \cap M)\). [2]
3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram. [3]

In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that \(\text{P}(W) = 0.14\), \(\text{P}(F) = 0.41\) and \(\text{P}(W \cap F) = 0.11\).

1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(W\) and \(F\) are independent. [2]
3. Find \(\text{P}(W|F)\) and explain what this probability represents. [3]

The table shows all the possible products of the scores on two fair four-sided dice. \includegraphics{figure_6}

1. Find the probability that the product of the two scores is less than 10. [1]
2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]