Exploring the Probability Distribution of the Sum of Coin Toss and Die Roll Outcomes
In this article, we delve into a classic probability problem involving a coin toss and a die roll, focusing on the random variable X, which represents the sum of these outcomes. We will determine the range space of X and calculate the probabilities associated with various values of X.
Introduction to the Problem
We begin with a simple experiment: a coin is tossed, and the outcome (Heads or Tails) is recorded as 1 or 0. Additionally, a standard six-sided die is rolled, with outcomes ranging from 1 to 6. The random variable X is then defined as the sum of the outcomes from the coin toss and the die roll.
Determining the Range Space of X
First, we list the possible outcomes for each event:
Coin Toss Outcomes: Heads (recorded as 1) Tails (recorded as 0) Die Roll Outcomes: 1 2 3 4 5 6Next, we calculate the possible values of X based on these outcomes:
If the coin shows Heads (1): 1 1 2 1 2 3 1 3 4 1 4 5 1 5 6 1 6 7 If the coin shows Tails (0): 0 1 1 0 2 2 0 3 3 0 4 4 0 5 5 0 6 6By combining both cases, we obtain the complete set of possible values for X:
1, 2, 3, 4, 5, 6, 7Thus, the range space of the random variable X is {1, 2, 3, 4, 5, 6, 7}.
Probability Calculations
To find the probabilities associated with these values, we calculate the probability of each combination resulting in a specific sum:
If X 2: Coin is Heads (1) and Die is 1: P (frac{1}{2} times frac{1}{6} frac{1}{12}) Coin is Tails (0) and Die is 2: P (frac{1}{2} times frac{1}{6} frac{1}{12})The total probability for X 2 is the sum of these individual probabilities:
P(X 2) (frac{1}{12} frac{1}{12} frac{2}{12} frac{1}{6})We can perform a similar analysis for other values of X, but the calculations are similar in nature.
Summary and Conclusion
In conclusion, the random variable X, which represents the sum of the outcomes of a coin toss and a die roll, has a range space of {1, 2, 3, 4, 5, 6, 7}. The probabilities associated with each value of X can be calculated by considering the independent probabilities of the outcomes of the coin and the die. This problem demonstrates the fundamental concepts of probability distribution and random variables in a practical and accessible manner.