Question

13 buttons in a bag 9 are white 4 are black if you take two buttons out what is the probability that you get pne of each colour

Answer 1

With or without replacement?

Answer 2

With replacement, you have a 9/13 chance of getting a white button, and a 4/13 chance of getting a black button.
The probability of getting a black and a white is equal two the product of the two probabilities:
9/13*4/13=36/169 or about 21.3%

Answer 3

Without replacement it gets a bit more complicated... There are now two ways you can get one of each: by choosing a white then a black, or by choosing a black then a white.

since there is no replacement, its a changing fraction problem (you are drawing from a different number of buttons on the second draw)

Probability of choosing one of each
= probability of black then white + probability of white then black
= 4/13 * 9/12 + 9/13 * 4/12
= 6/13 or about 46%

Answer 4

In both cases you need to use \({2 \choose 1}\) to account for black then white, and white then black.

Answer 5

Notice that \(9\times 4 + 9\times 9+4\times 4 \neq 13\times 13\)