Problem 46 - Algebra for College Students

Question

Two dice are tossed 300 times. How many times would you expect to get a double? 50

Step-by-Step Solution

Answer

Expect 50 doubles when tossing the dice 300 times.

Step 1: Understand the Problem

In this problem, we are tossing two dice 300 times and need to find the expected number of doubles, which means both dice show the same number. Each die has 6 faces, which range from 1 to 6.

Step 2: Identify Possible Doubles

When both dice are thrown, a double occurs if both dice show the same number. Possible doubles are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). This gives us a total of 6 possible doubles.

Step 3: Calculate the Total Possible Outcomes

Each die has 6 faces, so when two dice are thrown, the total number of possible outcomes is \(6 \times 6 = 36\).

Step 4: Calculate the Probability of a Double

The probability of rolling a double is the number of favorable outcomes for doubles divided by the total number of outcomes. Therefore, the probability \(P(\text{double}) = \frac{6}{36} = \frac{1}{6}\).

Step 5: Calculate Expected Number of Doubles

Using the probability from Step 4 and the fact that the dice are tossed 300 times, the expected number of doubles is calculated by multiplying the probability by the number of trials: \(E = 300 \times \frac{1}{6} = 50\).

Key Concepts

Expected Value

Expected value is a key concept in probability and statistics. It represents the average outcome if an experiment is repeated many times. In the context of our die-rolling experiment, it's important to calculate how many times we expect a certain outcome, such as rolling doubles, over a series of trials.

To find the expected value, you multiply the probability of success of a single event by the number of trials. Mathematically, you represent it as:

• Expected Value (E) = Probability of Event × Number of Trials

In the example, since the probability of getting a double (like rolling two ones) is \(\frac{1}{6}\) and there are 300 trials, you compute the expected value as follows: \(E = 300 \times \frac{1}{6} = 50\).

Expected value gives you an idea of what to anticipate in any probabilistic scenario, guiding decisions in uncertain environments.

Die Rolling Experiment

Die rolling experiments are classic examples in probability and are often used to explain probability concepts simply and clearly. This type of experiment boils down to observing outcomes when one or more dice are rolled.

Each die has 6 faces with numbers ranging from 1 to 6. This setup provides various potential outcomes each time the dice are rolled. In our exercise, two standard dice are used, expanding the possible outcomes from 6 per die to 36 combined.

• One die outcomes: 1, 2, 3, 4, 5, 6
• Two dice outcomes: all combinations from (1,1) to (6,6), totaling 36

Conducting die rolling experiments can help students grasp the basics of probability distributions, expected outcomes, and more complex concepts such as variance and standard deviation.

Outcome Probability

Probability measures the likelihood of a specific outcome occurring in any experiment. In our die-rolling context, outcome probability quantifies the chance of rolling doubles.

Each face of a die has an equal chance of being rolled. For doubles, where both dice show the same number, there are 6 favorable outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). Given the dice can result in 36 possible combinations, the probability of rolling a double is:

• Probability of a Double = \(\frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} = \frac{6}{36} = \frac{1}{6}\)

Understanding outcome probability is critical, as it forms the basis for calculating expected values and making predictions about future trials.

Independent Events

Independent events are a crucial concept in probability theory. Two events are independent if the outcome of one does not influence the outcome of the other.

In the die-rolling experiment, each roll of the dice is independent. This is because the outcome of one die does not affect the outcome of another. Thus, when calculating probabilities and expected values, it's assumed that each roll is a separate event with no dependence on previous rolls.

• Example of Independence: Rolling a \(3\) on the first die doesn't change the probability of rolling a double.

Recognizing independence in events allows for straightforward multiplication of probabilities in sequences of trials, underpinning the entire expected value calculation.