Question

there are 2 sets of balls numbered 1 through 19 placed in a bowl. if 2 balls are randomly chosen without replacement, find the probability that the ball have the same number. express your answer as a fraction.

Answer 1

Step-by-step explanation

Let's recall Laplace's law:

`P(\text{Choosing two balls with the same number})=\frac{\#\text{ outcomes where the two balls have the same number}}{\#\text{ of total combinations of choosing two balls randomly}}\text{.}`

Our task now is to find the numerator and the denominator of the quotient above.

For each number from 1 to 19, we have only one combination of two balls with that number. Then, we have 19 outcomes where the two chosen balls have the same number.

On the other hand, note that we have a total of 19*2=38 balls in the bowl, and 2 of them are chosen. Then, the total number of combinations is given by

`_(38)C_2=(38!)/((38-2)!\cdot2!)=(38!)/(36!\cdot2!)=\frac{38\cdot37\cdot\cancel{36!}}{\cancel{36!}\cdot(2\cdot1)}=(38\cdot37)/(2)=19\cdot37=703.`

Replacing these values in Laplace's law, we get

`P(\text{Choosing two balls with the same number})=(19)/(703).`Answer

The probability of choosing two balls with the same number is

`(19)/(703)\text{.}`