3 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 \(\mathrm { P } ( T ) = 0.55 , \mathrm { P } ( M ) = 0.33\) and \(\mathrm { P } ( T \mid M ) = 0.80\).

1. Use this information to show that the events \(T\) and \(M\) are not independent.
2. Find \(\mathrm { P } ( T \cap M )\).
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.
   \(425 \%\) of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
4. Find the probability that there are no plants with red flowers in the sample.
5. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers.