A probability problem by Varun Gupta

Probability Level 3

A problem is given to 3 people A,B and C. Probability that they solve the problem is 1 2 \frac{1}{2} , 2 3 \frac{2}{3} and 3 4 \frac{3}{4} respectively. What is the probability that problem is solved?

23 24 \frac{23}{24} 1 24 \frac{1}{24} 1 4 \frac{1}{4} 3 4 \frac{3}{4}

Varun Gupta by Varun Gupta , 23, India

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2 solutions

Noel Lo
May 11, 2015

Hi Muhammad Alif Aqsha, I am not 100% sure but there seems to be a mistake in your method. For the problem to be solved, there need not necessarily be EXACTLY one person who solves it. It can be 2 out of 3 or all 3 solving it as well.

Now to all viewers of this problem, what I did is to assume that NONE could solve it as that is the ONLY possibility for the problem to be unsolved. The problem will be solved as long as AT LEAST ONE person is able to solve it. So how do I find the chance that none can solve? Multiply the complementary probability of all three = ( 1 − 1 2 ) ( 1 − 2 3 ) ( 1 − 3 4 ) = 1 2 ( 1 3 ) ( 1 4 ) = 1 24 (1-\frac{1}{2})(1-\frac{2}{3})(1-\frac{3}{4}) = \frac{1}{2} (\frac{1}{3}) (\frac{1}{4}) = \frac{1}{24} . This gives us the probability that problem is unsolved. So probability that problem is solved = 1 − 1 24 = 23 24 1-\frac{1}{24} = \frac{23}{24}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

It asks for the probability that the problem is solved. It can be A, B or C who solves it. So, 1 2 + 2 3 + 3 4 = 23 24 \frac {1}{2}+\frac {2}{3}+\frac {3}{4}=\frac {23}{24}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Please check your calculation. 1 2 + 2 3 + 3 4 = 23 12 ≠ 23 24 \frac {1}{2}+\frac {2}{3}+\frac {3}{4} = \frac{23}{12} \neq \frac{23}{24} .

To solve this question, using probabilty - by complement is probably the easiest method. The probability that the problem is not solved is 1 2 × 1 3 × 1 4 = 1 24 \frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} = \frac{1}{24} , therefore the probability that the problem is solved is 1 − 1 24 = 23 24 1 - \frac{1}{24} = \boxed{\frac{23}{24}}

Pranshu Gaba - 6 years, 1 month ago

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My mistake, sorry. But I had no idea why I answered it correctly, as I made such terrible miscalculation.

Muhammad Alif Aqsha - 6 years, 1 month ago

Excellent!

Varun Gupta - 6 years, 1 month ago

plz check ur calculation, it is 23/12

Rajesh Soni - 6 years, 1 month ago

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Friend, calculation is alright. It is the wrong method by which u r solving! Its answer can be get by calculating 1 - none of them solve. So we get /frac{23}{24} .

Varun Gupta - 6 years, 1 month ago

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