Dice is a popular game in gambling casinos. Two dice are tossed, and various amounts are paid according to the outcome. In a certain game, if a four or eleven occurs on the first roll, the player wins. What is the probability of winning on the first roll?

To calculate the probability of winning on the first roll by rolling a four or eleven, we need to determine the number of favorable outcomes and the total number of possible outcomes.

In this case, we are using two dice, each with six faces numbered 1 through 6. So the total number of possible outcomes when rolling two dice is 6 multiplied by 6, which equals 36.

Now, let's find the favorable outcomes. A four can occur in three different ways: 1-3, 2-2, and 3-1. Similarly, an eleven can occur in two different ways: 5-6 and 6-5.

Therefore, there are 3 favorable outcomes for rolling a four and 2 favorable outcomes for rolling an eleven, making a total of 3 + 2 = 5 favorable outcomes.

The probability of winning on the first roll is given by the number of favorable outcomes divided by the number of possible outcomes. So, the probability is 5 divided by 36.

Calculating this fraction, we find that the probability of winning on the first roll by rolling a four or eleven is approximately 0.1389, or about 13.89%.