2.03b Probability diagrams: tree, Venn, sample space

Of the cars that are taken to a certain garage for an M.O.T. test, 87% pass. However, 2% of these have faults for which they should have been failed. 5% of the cars which fail are in fact roadworthy and should have passed. Using a tree diagram, or otherwise, calculate the probabilities that a car chosen at random

1. should have passed the test, regardless of whether it actually did or not, [4 marks]
2. failed the test, given that it should have passed. [3 marks]

The garage is told to improve its procedures. When it is inspected again a year later, it is found that the pass rate is still 87% overall and 2% of the cars passed have faults as before, but now 0.3% of the cars which should have passed are failed and \(x\)% of the cars which are failed should have passed.

1. Find the value of \(x\). [8 marks]

Among the families with two children in a large city, the probability that the elder child is a boy is \(\frac{5}{12}\) and the probability that the younger child is a boy is \(\frac{9}{16}\). The probability that the younger child is a girl, given that the elder child is a girl, is \(\frac{1}{4}\). One of the families is chosen at random. Using a tree diagram, or otherwise,

1. show that the probability that both children are boys is \(\frac{1}{8}\). [5 marks]

Find the probability that

1. one child is a boy and the other is a girl, [3 marks]
2. one child is a boy given that the other is a girl. [3 marks]

If three of the families are chosen at random,

1. find the probability that exactly two of the families have two boys. [3 marks]
2. State an assumption that you have made in answering part (d). [1 mark]

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]

Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_5} A day is selected at random. Find the probability that

1. the weather is wet and Andy travels by bus, [2]
2. Andy walks or travels by bike, [3]
3. the weather is dry given that Andy walks or travels by bike. [3]

A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.

• Reducing car use
• Insulating their homes
• Avoiding air travel

The numbers of citizens who have used each of these methods are shown in the Venn diagram. \includegraphics{figure_6} One of the citizens is selected at random.

1. Find the probability that this citizen
   1. has avoided air travel, [1]
   2. has used at least two of the three methods. [2]
2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. [2]

Three of the citizens are selected at random.

1. Find the probability that none of them have avoided air travel. [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]

Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities 0.2, 0.5 and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.

1. Draw a probability tree diagram to illustrate the outcomes. [3]
2. Find the probability that a randomly selected candidate is accepted. [2]
3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted. [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]

A netball team are in a league with three other teams from which one team will progress to the next stage of the competition. The team's coach estimates their chances of winning each of their three matches in the league to be 0.6, 0.5 and 0.3 respectively, and believes these probabilities to be independent of each other.

1. Show that the probability of the team winning exactly two of their three matches is 0.36 [4 marks]

Let the random variable \(W\) be the number of matches that the team win in the league.

1. Find the probability distribution of \(W\). [4 marks]
2. Find E\((W)\) and Var\((W)\). [6 marks]
3. Comment on the coach's assumption that the probabilities of success in each of the three matches are independent. [2 marks]

Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is • 0.7 if he won the previous set, • 0.4 if Alan won the previous set. It is not possible to draw a set.

1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets. [3]
2. Find the probability that Alan wins the match. [3]
3. Find the probability that the match ends after exactly four sets have been played. [2]

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]

Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_7} A day is selected at random. Find the probability that

1. the weather is wet and Andy travels by bus, [2]
2. Andy walks or travels by bike, [3]
3. the weather is dry given that Andy walks or travels by bike. [3]

A plane flies regularly between airports D and T with an intermediate stop at airport M. The time of the plane's departure from, or arrival at, each airport is classified as either early, on time, or late. On 90\% of flights, the plane departs from D on time, and on 10\% of flights, it departs from D late. Of those flights that depart from D on time, 65\% then depart from M on time and 35\% depart from M late. Of those flights that depart from D late, 15\% then depart from M on time and 85\% depart from M late. Any flight that departs from M on time has probability 0.25 of arriving at T early, probability 0.60 of arriving at T on time and probability 0.15 of arriving at T late. Any flight that departs from M late has probability 0.10 of arriving at T early, probability 0.20 of arriving at T on time and probability 0.70 of arriving at T late.

1. Represent this information by a tree diagram on which labels and percentages or probabilities are shown. [3 marks]
2. Hence, or otherwise, calculate the probability that the plane:
   1. arrives at T on time;
   2. arrives at T on time, given that it departed from D on time;
   3. does not arrive at T late, given that it departed from D on time;
   4. does not arrive at T late, given that it departed from M on time.
   [8 marks]
3. Three independent flights of the plane depart from D on time. Calculate the probability that two flights arrive at T on time and that one flight arrives at T early. [4 marks]

Andy and Bev are playing a game.

• The game consists of three points.
• On each point, P(Andy wins) = 0.4 and P(Bev wins) = 0.6.
• If one player wins two consecutive points, then they win the game, otherwise neither player wins.

1. Determine the probability of the following events.
   1. Andy wins the game. [2]
   2. Neither player wins the game. [3]

Andy and Bev now decide to play a match which consists of a series of games.

• In each game, if a player wins the game then they win the match.
• If neither player wins the game then the players play another game.

1. Determine the probability that Andy wins the match. [3]

Numbered balls are placed in bowls A, B and C In bowl A there are four balls numbered 1, 2, 3 and 7 In bowl B there are eight balls numbered 0, 0, 2, 3, 5, 6, 8 and 9 In bowl C there are nine balls numbered 0, 1, 1, 2, 3, 3, 3, 6 and 7 This information is shown in the diagram below. \includegraphics{figure_15} A three-digit number is generated using the following method: • a ball is selected at random from each bowl • the first digit of the number is the ball drawn from bowl A • the second digit of the number is the ball drawn from bowl B • the third digit of the number is the ball drawn from bowl C

1. Find the probability that the number generated is even. [1 mark]
2. Find the probability that the number generated is 703 [2 marks]
3. Find the probability that the number generated is divisible by 111 [2 marks]

The probabilities of events \(A\), \(B\) and \(C\) are related, as shown in the Venn diagram below. \includegraphics{figure_10} Find the value of \(x\). Circle your answer. [1 mark] \(0.11\) \quad \(0.46\) \quad \(0.54\) \quad \(0.89\)

A sample of 240 households were asked which, if any, of the following animals they own as pets: • cats (C) • dogs (D) • tortoises (T) The results are shown in the table below.

1. Represent this information by fully completing the Venn diagram below. [3 marks] \includegraphics{figure_16}
2. A household is chosen at random from the sample.
   1. Find the probability that the household owns a cat only. [1 mark]
   2. Find the probability that the household owns at least two of the three types of pet. [2 marks]
   3. Find the probability that the household owns a cat or a dog or both, given that the household does not own a tortoise. [2 marks]
3. Determine whether a household owning a cat and a household owning a tortoise are independent of each other. Fully justify your answer. [2 marks]

The shaded region on one of the Venn diagrams below represents \((A \cup C) \cap B\) Identify this Venn diagram. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_13}

A survey of university students revealed that

• 31\% have a part-time job but do not play competitive sport.
• 23\% play competitive sport but do not have a part-time job.
• 22\% do not play competitive sport and do not have a part-time job.

1. Show this information on a Venn diagram. [2]

A student is selected at random.

1. Determine the probability that the student plays competitive sport and has a part-time job. [2]

The Venn diagram shows the subjects studied by 40 sixth form students. \(F\) represents the set of students who study French, \(M\) represents the set of students who study Mathematics and \(D\) represents the set of students who study Drama. The diagram shows the number of students in each set. \includegraphics{figure_2}

1. Explain what \(M \cap D'\) means in this context. [1]
2. One of these students is chosen at random. Find the probability that this student studies
   1. exactly two of these subjects,
   2. Mathematics or French or both. [3]
3. Determine whether studying Mathematics and studying Drama are statistically independent for these students. [3]