Venn Diagrams and Tree Diagrams

Key Questions

• How do Venn diagrams and tree diagrams help you calculate probabilities?

  They can make you see whether you can ADD probabilities or have to multiply.

  Let's take the tree first. You throw three coins. What are the chances there are at least two heads? Draw the tree!
  First split: H 0.5 -- T 0.5 let's do the H to the left.
  Second splits: again H 0.5 -- T 0.5
  So H-H has a chance of 0.5 times 0.5 = 0.25 and we can stop left-left
  T-T is already a looser (0.25)
  We still have H-T and T-H each with a chance of 0.25
  Third throw: they both have a chance of 0.5 so:
  H-T-H = 0.5 x 0.25 = 0.125
  T-H-H = 0.5 x 0.25 = 0.125
  Adding up, we have 0.25 + 0.125 + 0.125 =0.5
  If we had wanted to know the chance of exactly two heads we would have had to go on with the left-left, because H-H-H would then be excluded.
  Summary: down a tree, you multiply -- at the ends you add.

  MeneerNask · 1 · Dec 22 2014

• In a certain department, 45 students are enrolled in a chemistry class, 37 students are enrolled in biology, and 16 students are enrolled in both. Assuming every student is in at least one of the courses, how many students are enrolled in exactly one course?
• A basketball team plays 60% of its games at home. The home court advantage is obvious, because the team wins 70% of its home games, but when they play away, they win only 35% of their games. Create a tree diagram to illustrate the paths for home and away games and the outcome of winning or losing for each type of game.
• A basketball team plays 60% of its games at home. The home court advantage is obvious, because the team wins 70% of its home games, but when they play away, they win only 35% of their games. What is the (conditional) probability of losing, GIVEN THAT the game was played away?
• A company with 500 employees wants to have its group health insurance premiums lowered. Here is one question from the questionnaire that every employee filled out and returned to the HR department: What have you done in the past year to improve your health? (circle all that apply) A. I have eaten healthier. B. I have exercised regularly. When the results were tabulated, the HR clerk reported that 140 people circled response A, 290 circled response B, and 50 people circled both responses A and B. What percent of company employees circled neither A nor B?
• A company with 500 employees wants to have its group health insurance premiums lowered. Here is one question from the questionnaire that every employee filled out and returned to the HR department: What have you done in the past year to improve your health? (circle all that apply) A. I have eaten healthier. B. I have exercised regularly. When the results were tabulated, the HR clerk reported that 140 people circled response A, 290 circled response B, and 50 people circled both responses A and B. Given that a randomly selected individual answered B, what is the probability that he or she also answered A?
• How do Venn diagrams and tree diagrams help you calculate probabilities?
• Please refer to the image here. ?
• Question #bd631
• A survey shows that 48% of the respondents like soccer, 66% like basketball, and 38% like hockey. If Meg likes basketball, what is the probability that she also likes soccer?
• What is the difference between mutually exclusive and independent events?
• If events are mutually exclusive are they independent?
• Three events A, B and C are defined in the sample space S. The events A and B are mutually exclusive and A and C are independent. What is #P(A|C)#?
• Three events A, B and C are defined in the sample space S. The events A and B are mutually exclusive and A and C are independent. What is #P(AuuB)#?
• Which two-way table contains the same information as the Venn diagram?
• For a custom car, there are 7 choices of wheel covers, 4 choices of stereos, 8 choices of colors, and 2 choices of windows. If you made a tree diagram to show all the possible outcomes, how many leaves would the tree diagram have?
• In a group of 50 patrons, 18 patrons like lattes, 16 patrons like espressos, and 8 patrons like both coffee drinks. How many patrons don't like either of the coffee drinks?