[Math] Probability, making a selection of 2 people from 50, with conditions

conditional probabilityprobability

A president and a treasurer are to be chosen from a club consisting of 50 people. How many different choices of officers are possible if
A will serve only if he is president;
B and C will serve together or not at all;
D and E will not serve together?
I need to construct its tree diagram and find total number of ways of selection with given constraints.

Best Answer

You construct the tree by deciding how many ways there are to fill the president position, and for each distinct way, how many ways there are to fill the treasurer position, based on who is willing and acceptable.

  • If A is president, who can serve as treasurer?
    • Only B and C will not serve as treasurer if A is president
    • Thus there are $47$ people who are willing and acceptable to serve as treasurer when A is president.
  • If B or C is president, who can serve as treasurer?
    • ...
  • If D or E is president, who can serve as treasurer?
    • ...
  • If any of the $45$ other people are president, who can serve to be treasurer?
    • ...

$$1\times 47 + 2 \times \underline\quad + 2\times \underline\quad + 45 \times \underline\quad$$

Fill in the blanks.

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