three people A,B,C set out to try to solve a puzzle

Question

From past experience it is thought that the probability of A solving it is 1/2 B solving it is 1/4 and C solving is 1/3. What is the probability that the puzzle will be solved.

Answer 1

To say "the puzzle will be solved" is the same as saying A will solve it, or B will solve it, or C will solve it, or any combination of people will solve it.  So that's the same as P(A or B or C).

 

The rule for the probability of event A or event B is:

 

P(A or B) = P(A) + P(B) - P(A and B)

 

You can extend this to cover any number of events, not just 2.  For 3 events the rule is:

 

P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C)

 

When the events are independent, which is the case here because one person's solving the puzzle doesn't affect another person's solving it, the "and" probabilities are equal to products:

 

P(A or B or C) = P(A) + P(B) + P(C) - P(A) P(B) - P(A) P(C) - P(B) P(C) + P(A) P(B) P(C)

 

So you get the following:

 

P(A or B or C) = 1/2 + 1/4 + 1/3 - (1/2)(1/4) - (1/2)(1/3) - (1/4)(1/3) + (1/2)(1/4)(1/3) =

1/2 + 1/4 + 1/3 - 1/8 - 1/6 - 1/12 + 1/24 =

12/24 + 6/24 + 8/24 - 3/24 - 4/24 - 2/24 + 1/24 =

18/24 =

3/4

 

Answer 2

The easiest way to solve this problem is to:

 

Find the probability that the problem will not be solved.

 

Subtract this probability from 1, to get the probability that it will be solved.

 

Probability that A will not solve the problem:  1-1/2 = 1/2

 

Probability that B will not solve the problem:  1-1/4 = 3/4

 

Probability that C will not solve the problem:  1-1/3 = 2/3

 

Probability that the problem will not be solved is:  Prob(not A) x Prob(not B) x Prob(not C)

 

Probability problem not solved: (1/2)(3/4)(2/3) = 1/4

 

Probability that problem will be solved:  1-1/4 = 3/4

 

Answer 3

Another way to think about this is to calculate the probability that the puzzle will not be solved and then use the formula:

P(puzzle will be solved) = 1-P(puzzle will not be solved)

 

Now, puzzle will not be solved if none of the three people can solve it.

P(A will not solve the puzzle) = 1-1/2 = 1/2

P(B will not solve the puzzle) = 1-1/4 = 3/4

P(C will not solve the puzzle) = 1-1/3 = 2/3

 

P(puzzle will not be solved) = P(A will not solve it)*P(B will not solve it)*P(C will not solve it)

                                       = (1/2)*(3/4)*(2/3)

                                       = 1/4

Therefore,

P(puzzle will be solved) = 1-P(puzzle will not be solved)
                                 = 1-1/4 = 3/4