# Thirty-three cards numbered from 1 through 33 are placed into a box and two cards are selected without replacement. Find the probability that both numbers selected are odd, given that their sum is even.

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Thirty-three cards numbered from 1 through 33 are placed into a box and two cards are selected without replacement. Find the probability that both numbers selected are odd, given that their sum is even.

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Step 1

Given that thirty three cards are numbered from 1 to 33 and placed in a box.

The total number of odd numbered cards = 17

The total number of even numbered cards = 16

The sum of numbers on two cards is even only when both are even numbered or both are odd numbered.

total number of ways of picking two odd cards without replacement = 17C116C1 = 17*16 = 272

Total number of ways of picking two even numbers without replacement = 16C1*15C1 = 16*15 = 240

Step 2

The total number of ways of picking two cards from 33 cards = 33C1* 32C1 = 33*32 = 1056.

Let the event of drawing two cards with sum of numbers is even be E and the event of drawing two odd numbered cards be A and event of drawing two even numbered cards be B.

Probability of drawing two odd numbered card...

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