Independence test with conditional probability

Questions that provide P(A|B) or require calculating it, and test independence by checking if P(A|B) = P(A) or equivalent conditional probability relationships.

2 questions

OCR MEI S1 Q4
4 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.
  2. Find \(\mathrm { P } ( L \cap R )\).
  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.
OCR MEI S1 Q3
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.