Question 1163269
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In a cafeteria students may order any combination of Chips, Sandwiches and Cold  drink.  
The  probability  that  a  student  chooses  a  cold  drink  is  0.45, 
sandwiches and  chips 0.19, cold  drink  and sandwiches 0.15,  cold  drink and chips 0.25, 
cold drink or sandwiches 0.6, cold drink or chips 0.84, 
cold drink or chips or sandwiches 0.9. 
Find the probability that a student chooses all the three. 
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In my solution, I will use the letters D, S and C  to coding drinks, sandwiches and chips, respectively.


So, we have an Universal set  U  of all possible combinations, and P(U) = 1.


We also have  given

    P(D) = 0.45
     
    P(S &#8745; C) = 0.19

    P(D &#8745; S) = 0.15

    P(D &#8745; C) = 0.25

    P(D U S) = 0.6

    P(D U C) = 0.84

    P(D U C U S) = 0.9.


There is a REMARKABLE formula in elementary probability theory

    P(D U C U S) = P(D) + P(C) + P(S) - P(D &#8745; S) - P(D &#8745; C) - P(C &#8745; S) + P(D &#8745; C &#8745; S)    (1)

which is valid for any subsets (sets of events) of the Universal set.


In this formula, many terms are given, but not all. The terms P(S) and P(D) are not given.

Had I knew these two terms, I would be in position to calculate the value of P(D &#8745; C &#8745; S) from the formula (1) MOMENTARILY.



So, my goal now is to find these terms   P(S) and P(D).



To find P(S),  I will use another basic formula of the elementary probability theory  P(D U S) = P(D) + P(S) - P(D &#8745; S).

By substituting all given values, I get  0.6 = 0.45 + P(S) - 0.15;  it implies  P(S) = 0.6 - 0.45 + 0.15 = 0.3.



To find P(C),  I will use similar basic formula                                       P(D U C) = P(D) + P(C) - P(D &#8745; C).

By substituting all given values, I get  0.84 = 0.45 + P(C) - 0.25;  it implies  P(C) = 0.84 - 0.45 + 0.25 = 0.64.



Now I substitute all the given and found values into the formula (1)

    0.9 = 0.45 + 0.3 + 0.64 - 0.19 - 0.15 - 0.2 + P(D &#8745; C &#8745; S),


which gives me the <U>ANSWER</U>

    P(D &#8745; C &#8745; S) = 0.9 - 0.45 - 0.3 - 0.64 + 0.19 + 0.15 + 0.2 = 0.1.
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Solved.


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I got the same answer as Edwin.


As usual in such problems, Edwin believes in Venn diagrams and auxiliary systems of linear equations in 8 unknowns.


I, in opposite, believe in simple basic formulas of the elementary probability theory.


Working together, we provide the reader with all possible ways to attack the problem.



<pre>
    My credo, as always, is this : the solution should be as simple as possible,

    or, if you want, as adequately simple as a problem is.
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