A box contains 4 black hats, 3 brown hats, and 2 white hats. What is the probability that the first hat chosen is brown and the second hat chosen is white?

To determine the probability of choosing a brown hat first and a white hat second from a box containing 4 black hats, 3 brown hats, and 2 white hats, we must first find the total number of hats. This total is calculated as follows:

• Number of black hats: 4

• Number of brown hats: 3

• Number of white hats: 2

Adding these together gives a total of 4 + 3 + 2 = 9 hats.

Now, let's find the probability of choosing a brown hat first. The number of favorable outcomes (choosing a brown hat) is 3 (the number of brown hats) out of 9 total hats. Thus, the probability of choosing a brown hat first is:

[

P(\text{brown first}) = \frac{3}{9} = \frac{1}{3}

]

Next, after removing one brown hat, there will be a total of 8 hats remaining in the box (since one brown hat was taken). The number of white hats remains at 2.

Now, we calculate the probability of then choosing a white hat:

[

P(\text{white second | brown first}) = \frac{2