Question

a coin is tossed two times. the set of outcomes is given as \\(s = \\{\text{hh}, \text{ht}, \text{th}, \text{tt}\\}\\).
let \\(x\\) represent the number of times heads occurs.

complete the table.
\\(a = \\)
\\(b = \\)
\\(c = \\)

Explanation:

Identify the sample space and outcomes

Using the Sample Space knowledge point
The sample space \(S\) for tossing a coin two times is given as:

$$ S = \{\text{HH}, \text{HT}, \text{TH}, \text{TT}\} $$

The total number of equally likely outcomes is:

$$ N(S) = 4 $$

Determine favorable outcomes for each value of X

Using the Favorable Outcomes knowledge point
Let \(X\) represent the number of times heads occurs. We analyze the outcomes for each value of \(X\):

• For \(X = 0\) (no heads): The favorable outcome is \(\{\text{TT}\}\). The number of favorable outcomes is \(1\).
• For \(X = 1\) (one head): The favorable outcomes are \(\{\text{HT}, \text{TH}\}\). The number of favorable outcomes is \(2\).
• For \(X = 2\) (two heads): The favorable outcome is \(\{\text{HH}\}\). The number of favorable outcomes is \(1\).

Calculate the probabilities

Using the Theoretical Probability knowledge point
We calculate the probability \(P_X(x) = \frac{\text{Number of favorable outcomes}}{N(S)}\) for each value:

• For \(X = 0\):

$$ a = P_X(0) = \frac{1}{4} = 0.25 $$

• For \(X = 1\):

$$ b = P_X(1) = \frac{2}{4} = \frac{1}{2} = 0.5 $$

• For \(X = 2\):

$$ c = P_X(2) = \frac{1}{4} = 0.25 $$

Answer:

Variable	Value
\(b\)	\(0.5\) (or \(\frac{1}{2}\))
\(c\)	\(0.25\) (or \(\frac{1}{4}\))