Venn diagram with two events

5 Each day the probability that Ashwin wears a tie is 0.2 . The probability that he wears a jacket is 0.4 . If he wears a jacket, the probability that he wears a tie is 0.3 .

1. Find the probability that, on a randomly selected day, Ashwin wears a jacket and a tie.
2. Draw a Venn diagram, using one circle for the event 'wears a jacket' and one circle for the event 'wears a tie'. Your diagram should include the probability for each region.
3. Using your Venn diagram, or otherwise, find the probability that, on a randomly selected day, Ashwin
   (A) wears either a jacket or a tie (or both),
   (B) wears no tie or no jacket (or wears neither).

5 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 \(\mathrm { P } ( W ) = 0.14 , \mathrm { P } ( F ) = 0.41\) and \(\mathrm { P } ( W \cap F ) = 0.11\).

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.
2. Determine whether the events \(W\) and \(F\) are independent.
3. Find \(\mathrm { P } ( W \mid F )\) and explain what this probability represents.

4 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 \(\mathrm { P } ( G ) = 0.24 , \mathrm { P } ( R ) = 0.13\) and \(\mathrm { P } ( G \cap R ) = 0.06\).

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.
2. Determine whether the events \(G\) and \(R\) are independent.
3. Find \(\mathrm { P } ( R \mid G )\).

3 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 \(\mathrm { P } ( G ) = 0.24 , \mathrm { P } ( R ) = 0.13\) and \(\mathrm { P } ( G \cap R ) = 0.06\).

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.
2. Determine whether the events \(G\) and \(R\) are independent.
3. Find \(\mathrm { P } ( R \mid G )\).

16 Research conducted by social scientists has shown that \(16 \%\) of young adults smoke cigarettes. Two young adults are selected at random.

1. Determine the probability that one smokes cigarettes and the other doesn't. The same research has also shown that
   • 75\% of young adults drink alcohol.
   • \(66 \%\) of young adults drink alcohol, but do not smoke cigarettes.
   • Determine the probability that a young adult selected at random does smoke cigarettes, but does not drink alcohol.
   • A young adult who drinks alcohol is selected at random. Determine the probability that this young adult smokes cigarettes.
   • Using your answer to part (c), explain whether the event that a young adult selected at random smokes cigarettes is independent of the event that a young adult selected at random drinks alcohol.

6 Twins Alec and Eric are members of the same local cricket club and play for the club’s under 18 team. The probability that Alec is selected to play in any particular game is 0.85 .
The probability that Eric is selected to play in any particular game is 0.60 .
The probability that both Alec and Eric are selected to play in any particular game is 0.55 .

1. By using a table, or otherwise:
   1. show that the probability that neither twin is selected for a particular game is 0.10 ;
   2. find the probability that at least one of the twins is selected for a particular game;
   3. find the probability that exactly one of the twins is selected for a particular game.
2. The probability that the twins' younger brother, Cedric, is selected for a particular game is:
   0.30 given that both of the twins have been selected;
   0.75 given that exactly one of the twins has been selected;
   0.40 given that neither of the twins has been selected. Calculate the probability that, for a particular game:
   1. all three brothers are selected;
   2. at least two of the three brothers are selected.
      (6 marks)

3. In a study of 120 pet-owners it was found that 57 owned at least one dog and of these 16 also owned at least one cat. There were 35 people in the group who didn't own any cats or dogs. As an incentive to take part in the study, one participant is chosen at random to win a year's free supply of pet food. Find the probability that the winner of this prize

1. owns a dog but does not own a cat,
2. owns a cat,
3. does not own a cat given that they do not own a dog.

1. Each day Anna drives to work.

• R is the event that it is raining.
• \(L\) is the event that Anna arrives at work late.

You are given that \(\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25\) and \(\mathrm { P } ( R \cup L ) = 0.41\).
i. Draw a Venn diagram showing the events \(R\) and \(L\). Fill in the probability corresponding to each of the four regions of your diagram.
ii. Determine whether the events \(R\) and \(L\) are independent.
iii. Find \(\mathrm { P } ( L \mid R )\)