# A bowl contains 25 balls numbered 1 to 25. A ball is drawn and its number is noted. Without replacing the first ball, another ball is drawn. The probability that the numbers on both balls are odd numbers is ?

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## Question:

A bowl contains 25 balls numbered 1 to 25. A ball is drawn and its number is noted. Without replacing the first ball, another ball is drawn. The probability that the numbers on both balls are odd numbers is ?

## Answer:

Using the hypergeometric distribution, it is found that there is a 0.26 = 26% probability that the numbers on both balls are odd numbers.

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The balls are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

The probability of x successes is given by the following formula:

In which:

• x is the number of successes.
• N is the size of the population.
• n is the size of the sample.
• k is the total number of desired outcomes.

Combinations formula:

is the number of different combinations of x objects from a set of n elements, given by the following formula.

In this question:

• 25 balls means that
• From 1 to 25, there are 13 odd numbers, thus
• 2 balls are chosen, which means that
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The probability that the numbers on both balls are odd numbers is ?

This is P(X = 2). Thus

0.26 = 26% probability that the numbers on both balls are odd numbers.

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