Challenge Problem Showdown – February 21st, 2012

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Here is this week’s problem:

Two integers between 1 and 100, inclusive, each randomly and independently chosen, are either added or multiplied, with an equal chance of either operation. What is the probability that the result is even?

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By: mmapplebeck | 2012/02/21 | Challenge Problem | Trackback | Comments [RSS 2.0]

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  1. Kirin
    Comment by Kirin | 2012/03/04 at 19:21:10

    x, y are in {1-100}. 50 are even, 50 are odd.
    p(x is even) = 50/100 = 1/2.

    Event1: pick 2 numbers:
    chances of (even, odd), (even, even), (odd, even):
    p(even, odd) = p(even) * p(odd) = 1/2 * 1/2 = 1/4.
    3 combinations so 3*1/4 = 3/4.

    Event2: pick either addition or multiplication. x+y or x*y. This is half. Either you add the numbers or you multiply the two numbers.

    Chances of getting even: (pick right numbers) (pick right calculation method)
    p(even, odd) & p(even * odd) = 1/4*1/2=1/8
    p(even, even) & p(either add or mult will work) = 1/4 * 2/2 = 1/4
    p(odd, odd) & p(odd + odd) = 1/4 * 1/2=1/8

    so prob getting even given we pick any two numbers and randomly use addition or multiplication is: 1/8+1/4+1/8 = 2/4 = 1/2.

    alternative:
    odd*odd=odd, odd+even=odd all other pairings are even.
    1-p(odd) = even.

    Event1: probability of picking odd,odd or odd,even.
    p(odd, odd)=1/2*1/2=1/4
    p(odd,even)=1/2*1/2=1/4
    both: 1/4+1/4=1/2

    Event2: probability of picking addition or multiplication
    p(1 out of 2 choices)= 1/2

    chances of getting odd:
    p(odd1,odd2) * p(multiplication) = 1/4 * 1/2 = 1/8
    p(odd,even) * p(addition) = 1/4 * 1/2 = 1/8
    chances of odd= 1/8+1/8 = 1/4.
    chances of even = 1 – odd = 1 – 1/4 = 3/4.

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  2. Kirin
    Comment by Kirin | 2012/03/04 at 19:47:38

    There is a 50/50 chance of picking odd or even numbers. So the probability is mostly whether multiplication and addition favor even or odd. It favors even. we know that:

    even * even = even, even+even=even.
    odd*odd = odd, odd+odd=even.
    even*odd = even, even+odd=odd.
    4/6 is even.

    since we are choosing randomly and independently, include odd*even though this is same as even*odd.
    odd*even = even, odd+even=odd.
    so 5 evens out of 8 possibilities.
    answer 5/8.


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  1. Kirin
    Comment by Kirin | 2012/03/04 at 19:21:10

    x, y are in {1-100}. 50 are even, 50 are odd.
    p(x is even) = 50/100 = 1/2.

    Event1: pick 2 numbers:
    chances of (even, odd), (even, even), (odd, even):
    p(even, odd) = p(even) * p(odd) = 1/2 * 1/2 = 1/4.
    3 combinations so 3*1/4 = 3/4.

    Event2: pick either addition or multiplication. x+y or x*y. This is half. Either you add the numbers or you multiply the two numbers.

    Chances of getting even: (pick right numbers) (pick right calculation method)
    p(even, odd) & p(even * odd) = 1/4*1/2=1/8
    p(even, even) & p(either add or mult will work) = 1/4 * 2/2 = 1/4
    p(odd, odd) & p(odd + odd) = 1/4 * 1/2=1/8

    so prob getting even given we pick any two numbers and randomly use addition or multiplication is: 1/8+1/4+1/8 = 2/4 = 1/2.

    alternative:
    odd*odd=odd, odd+even=odd all other pairings are even.
    1-p(odd) = even.

    Event1: probability of picking odd,odd or odd,even.
    p(odd, odd)=1/2*1/2=1/4
    p(odd,even)=1/2*1/2=1/4
    both: 1/4+1/4=1/2

    Event2: probability of picking addition or multiplication
    p(1 out of 2 choices)= 1/2

    chances of getting odd:
    p(odd1,odd2) * p(multiplication) = 1/4 * 1/2 = 1/8
    p(odd,even) * p(addition) = 1/4 * 1/2 = 1/8
    chances of odd= 1/8+1/8 = 1/4.
    chances of even = 1 – odd = 1 – 1/4 = 3/4.

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    1. Kirin
      Comment by Kirin | 2012/03/04 at 19:47:38

      There is a 50/50 chance of picking odd or even numbers. So the probability is mostly whether multiplication and addition favor even or odd. It favors even. we know that:

      even * even = even, even+even=even.
      odd*odd = odd, odd+odd=even.
      even*odd = even, even+odd=odd.
      4/6 is even.

      since we are choosing randomly and independently, include odd*even though this is same as even*odd.
      odd*even = even, odd+even=odd.
      so 5 evens out of 8 possibilities.
      answer 5/8.

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