A box contains 15 blue balls and 45 black balls. If 2 balls are selected randomly, without replacement, the probability of an outcome in which the first selected is a blue ball and the second selected is a black ball, is _____

The correct option is 45/236.

Let us find the probability step-by-step.
First, let's determine the total number of balls in the box:
Number of blue balls = 15
Number of black balls = 45
Total number of balls = 15 + 45 = 60.

We want to find the probability of the event where the first ball selected is blue and the second ball selected is black, without replacement.

Step 1: Probability of selecting a blue ball first
In the beginning, there are 15 blue balls out of a total of 60 balls.
Therefore, the probability of selecting a blue ball on the first draw is:

$$P\left(\text{First is Blue}\right)=\frac{15}{60}=\frac{1}{4}$$

Step 2: Probability of selecting a black ball second (without replacement)
Since the first ball was selected without replacement, we now have one less ball in the box.
Remaining total number of balls = 60 - 1 = 59.
The number of black balls remains unchanged at 45 because the first ball drawn was blue.
Thus, the conditional probability of selecting a black ball on the second draw, given that the first ball was blue, is:

$$P\left(\text{Second is Black}\right|\text{First is Blue})=\frac{45}{59}$$

Step 3: Calculating the combined probability
By the multiplication rule of probability, the probability of both events occurring is the product of their individual probabilities:

$$P\left(\text{Blue first and Black second}\right)=P\left(\text{First is Blue}\right)\times P\left(\text{Second is Black}\right|\text{First is Blue})$$

$$P\left(\text{Blue first and Black second}\right)=\frac{15}{60}\times \frac{45}{59}$$

$$P\left(\text{Blue first and Black second}\right)=\frac{1}{4}\times \frac{45}{59}$$

$$P\left(\text{Blue first and Black second}\right)=\frac{1\times 45}{4\times 59}=\frac{45}{236}$$

Thus, the probability of selecting a blue ball first and a black ball second is 45/236.