- The sample space is \(S = \{HH, HT, TH, TT\}\), containing \(4\) equally likely outcomes.
- For \(X = 0\), the only outcome is \(TT\), so \(a = \frac{1}{4}\).
- For \(X = 1\), the outcomes are \(HT\) and \(TH\), so \(b = \frac{2}{4} = \frac{1}{2}\).
- For \(X = 2\), the only outcome is \(HH\), so \(c = \frac{1}{4}\).
- Final values: \(a = \boxed{\frac{1}{4}}, b = \boxed{\frac{1}{2}}, c = \boxed{\frac{1}{4}}\).
Explanation
- Define the sample space and random variable
To find the probability distribution of the random variable \(X\), representing the number of heads in two fair coin tosses, we first define the sample space \(S\). The sample space is the set of all possible outcomes when tossing a coin twice: \(S = \{HH, HT, TH, TT\}\). Since each of these \(4\) outcomes is equally likely, each has a probability of \(\frac{1}{4}\).
- Calculate the probabilities for each value of X
Next, we look at the possible values for \(X\), the number of heads:
- For \(X = 0\): The only outcome is \(\{TT\}\). There is \(1\) such outcome. Thus, \(a = P(X=0) = \frac{1}{4}\).
- For \(X = 1\): The outcomes are \(\{HT, TH\}\). There are \(2\) such outcomes. Thus, \(b = P(X=1) = \frac{2}{4} = \frac{1}{2}\).
- For \(X = 2\): The only outcome is \(\{HH\}\). There is \(1\) such outcome. Thus, \(c = P(X=2) = \frac{1}{4}\).
- Verify results and present the final answer
Finally, we verify that the probabilities sum to \(1\):
\(a + b + c = \frac{1}{4} + \frac{2}{4} + \frac{1}{4} = \frac{4}{4} = 1\).
This confirms our calculations are correct. The table is filled as follows:
- \(a = \frac{1}{4}\)
- \(b = \frac{1}{2}\)
- \(c = \frac{1}{4}\)
Examples
In many fields like statistics, insurance, or game design, calculating the probability of discrete events is fundamental. For example, if you are designing a board game that uses two coins to move pieces, knowing that you have a 50% chance of getting exactly one head helps you set the rules for the game's movement, ensuring a balanced experience for all players. Applying these probability principles allows you to make informed predictions about outcomes in any system driven by random processes.