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Math Help - [SOLVED] cinema problem

  1. #1
    Junior Member JoanF's Avatar
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    [SOLVED] cinema problem

    4 friends go to the cinema and reserve 4 tickets. The seats are already assigned to them. When they arrive to the cinema, they sit at random. The probability of 1 stay in its place is greater or minor than or equal to the probability of 2 sit in its place?
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  2. #2
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    Hello, JoanF!

    4 friends go to the cinema and reserve 4 tickets.
    The seats are already assigned to them.
    When they arrive to the cinema, they sit at random.

    The probability that one is in the right seat is:
    . . (1) greater than
    . . (2) less than
    . . (3) equal to
    the probability that two are in the right seats?
    There are 4! = 24 possible seatings.

    Suppose the four friends are: . \{A,B,C,D\}
    . . and their assigned seats are in this order: . A\:B\:C\:D


    One in the right seat.

    There are 4 choices for the One.
    . . Suppose it is A.

    Then there are 2 ways for the others to sit in the wrong seats:
    . . A\:C\:D\:B\;\;\text{ and }\;\;A\:D\:B\:C

    Hence, there are: . 4 \times 2 \:=\:8 ways.

    . . P(\text{one in the right seat}) \:=\:\frac{8}{24}\:=\:\frac{1}{3}


    Two in the right seats.

    There are: . {4\choose2} = {\color{blue}6} choices for the Two.
    . . Suppose they are A and B.

    Then there is 1 way for the other two to be in the wrong seats:
    . . A\:B\:D\:C

    Hence, there are: . 6 \times 1 \:=\:6 ways.

    . . P(\text{two in the right seats}) \:=\:\frac{6}{24}\:=\:\frac{1}{4}


    Therefore: . P(\text{one in right seat}) \;\;{\color{red}>}\;\;P(\text{two in the right seats})

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