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).

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 \(\mathrm { P } ( R \mid G )\).

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

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.

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]