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Probability

by BTGmoderatorRO » Sun Dec 17, 2017 5:26 am
Melissa enters a charity raffle in which bags of candy are being given away as prizes. She will be randomly assigned a number from the set of positive integers from 1 to 500 and receive a bag filled with that many pieces of candy. She decides that if the number of pieces of candy she receives is such that she can distribute the candy equally among her three children OR such that she can distribute the candy equally among her children, herself, and her husband, then she will keep it. In all other cases, she will give the candy to her neighbor if only if it can be distributed equally among the seven members of her neighbor's family. What is the probability that Melissa gives the bag of candy to her neighbor?

A. 0.068
B. 0.076
C. 0.096
D. 0.114
E. 0.142

OA is B

Pls can any expert give me a mathematical apptroach to this question? Thanks a lot.
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by Jay@ManhattanReview » Mon Dec 18, 2017 1:04 am
Roland2rule wrote:Melissa enters a charity raffle in which bags of candy are being given away as prizes. She will be randomly assigned a number from the set of positive integers from 1 to 500 and receive a bag filled with that many pieces of candy. She decides that if the number of pieces of candy she receives is such that she can distribute the candy equally among her three children OR such that she can distribute the candy equally among her children, herself, and her husband, then she will keep it. In all other cases, she will give the candy to her neighbor if only if it can be distributed equally among the seven members of her neighbor's family. What is the probability that Melissa gives the bag of candy to her neighbor?

A. 0.068
B. 0.076
C. 0.096
D. 0.114
E. 0.142

OA is B

Pls can any expert give me a mathematical apptroach to this question? Thanks a lot.
Melissa will give the candy to her neighbor if the number of candies is a multiple of 7.

She will not give candies to her neighbor if the number of candies is a multiple of 3 or a multiple of 5. Therefore the cases in which she will give the candy to her neighbor is given by: Cases in which the number of candies she gets is a multiple of 7 but not a multiple of 3 or a multiple of 5.

The number of multiples of 7 between 1 and 500 = 71;

LCM of 3 and 7 = 21. The number of multiples of 21 between 1 and 500 = 23; she will not give to neighbor
LCM of 5 and 7 = 35. The number of multiples of 35 between 1 and 500 = 14; she will not give to neighbor
LCM of 3, 5, and 7 = 105. Total The number of multiples of 105 between 1 and 500 = 4; we must deduct 4 numbers of multiples as they are counted twice, once in 23 and once in 14.

So the number of multiples of 7 between 1 and 500 which are not multiples of 3 or 5 = 71 - (23 + 14 - 4) = 38

The probability that Melissa gives the bag of candy to her neighbor = 38/500 = 0.076

The correct answer: B

Hope this helps!

-Jay
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