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Chapter 11: Problem 88

Nine cards numbered from 1 through 9 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.

Short Answer

Expert verified
The required probability is 1.

Step by step solution

01

Determine Total Possible Outcomes

Firstly, consider the numbers 1 through 9. Out of these numbers, 5 are odd: 1, 3, 5, 7, and 9. The total number of ways to select 2 cards from these 5 odd numbers can be calculated using the combination formula \(\text{C}(n, r) = \frac{n!}{r!(n - r)!}\), where n is the total number of items, and r is the number of items to choose. So the total possible outcomes is \(\text{C}(5, 2) = 10\).
02

Determine Event of interest

The event of interest is that both numbers selected are odd. As already established, any two odd numbers will sum up to an even number. So, the event of both numbers selected being odd is equivalent to the event of their sum being even when choosing from a set of odd numbers. Therefore, the number of favourable outcomes is the same as the total outcomes which is 10.
03

Calculate the Conditional Probability

Since the number of favourable outcomes is the same as the total possible outcomes, the conditional probability is \(\frac{10}{10} = 1\) which is our required probability.

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