Question

For the experiments, list the simple events in the sample space, assign probabilities to the simple events, and find the required probabilities. A fair die is tossed twice. What is the probability that the sum of the two dice is \(11 ?\)

Short answer

Answer: The probability that the sum of two dice is 11 is \(\boxed{\frac{1}{18}}\).

Step-by-step solution

List the sample space

When a fair die is tossed twice, the sample space consists of all ordered pairs of numbers from 1 to 6. So, let's list all the possible outcomes (36 in total): $(1,1),(1,2),(1,3),(1,4),(1,5),(1,6), \\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6), \\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6), \\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6), \\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6), \\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)$

Assign probabilities to simple events

Since the die is fair, each outcome has an equal probability of occurring. Since there are 36 possible outcomes, the probability of each simple event is: \(P(simple\ event)=\frac{1}{36}\)

Finding the favorable outcomes

Now, we need to find the outcomes where the sum of the two dice is 11. The outcomes that satisfy this condition are: \((5,6),(6,5)\)

Calculate the probability

Since we have 2 favorable outcomes, and the probability of each simple event is \(\frac{1}{36}\), the probability that the sum of the two dice is 11 is: \(P(sum\ of\ two\ dice\ is\ 11)=2 \times \frac{1}{36} = \frac{2}{36} = \boxed{\frac{1}{18}}\)

Key concepts

Sample Space

When dealing with probability, the concept of a "sample space" is crucial. It represents all possible outcomes in a given experiment. By understanding the sample space, we can better predict the likelihood of various events.
A common probability problem involves rolling a fair die—let's say, doing it twice. The sample space for this experiment includes all the ordered pairs of numbers you could roll. Each die has six sides, numbered from 1 to 6, so if you roll two dice, the possibilities multiply. Specifically, you get 36 combinations.
Listed out, these combinations start with (1,1), then (1,2), all the way through to (6,6). This complete list of ordered pairs forms the sample space. It gives a comprehensive view of every outcome we might expect from rolling two dice.

Probability Calculation

Once we have our sample space, we need to assign probabilities to each simple event within it. In probability theory, a "simple event" is each individual outcome that can occur in an experiment. If we're dealing with a fair die, each simple event in our sample space has an equal likelihood of happening.
For the two-dice example, there are 36 possible outcomes. Thus, the probability of any one of these outcomes occurring is \( \frac{1}{36} \). This equal distribution is because the die is fair, meaning each face has an equal chance of being the result of a roll.
Understanding how we assign probabilities in fair and unbiased situations is fundamental to calculating probabilities in more complex scenarios.

Favorable Outcomes

Identifying "favorable outcomes" requires us to focus on specific events we are interested in predicting. In our two-dice example, we want to find the likelihood of the dice summing to 11. This event isn't just about knowing the sample space; we need to pinpoint exactly which outcomes meet the criteria.
The event "sum of the two dice is 11" only occurs in two situations: (5,6) and (6,5). These are our favorable outcomes because they satisfy the condition we're investigating.
Once we know the favorable outcomes, we can determine their probability. Since each outcome has a probability of \( \frac{1}{36} \), we multiply this probability by the number of favorable outcomes, which is 2 in this case. So, the probability of rolling a sum of 11 with two dice is \( 2 \times \frac{1}{36} = \frac{1}{18} \).
Grasping the concept of "favorable outcomes" helps in solving many probability theory problems, as it focuses our analysis on specific results that meet the conditions of our inquiry.