Understanding the Dice Roll Experiment

We are considering a random experiment involving rolling a single fair die. The possible outcomes (sample space) are:

$S = {1, 2, 3, 4, 5, 6}$

The total number of possible outcomes is $6$.

Defining the Events

Let's define the two events mentioned in the question:

• Event A: Getting an odd number. The outcomes for this event are ${1, 3, 5}$. The number of outcomes is $3$.
• Event B: Getting a number less than 4. The outcomes for this event are ${1, 2, 3}$. The number of outcomes is $3$.

Calculating Probability of Union of Events

We need to find the probability of Event A OR Event B occurring, which is represented as $P(A \cup B)$.

The formula for the probability of the union of two events is:

$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $

First, let's calculate the individual probabilities:

• Probability of Event A (getting an odd number):

  $ P(A) = \frac{\text{Number of odd outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} $

• Probability of Event B (getting a number less than 4):

  $ P(B) = \frac{\text{Number of outcomes < 4}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} $

Next, we need to find the intersection of A and B ($A \cap B$), which means the outcomes that are BOTH odd AND less than 4.

• Outcomes in $A \cap B$: ${1, 3}$. The number of outcomes is $2$.
• Probability of the intersection ($A \cap B$):

  $ P(A \cap B) = \frac{\text{Number of outcomes in A and B}}{\text{Total outcomes}} = \frac{2}{6} = \frac{1}{3} $

Now, substitute these values into the union formula:

$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $

$ P(A \cup B) = \frac{1}{2} + \frac{1}{2} - \frac{1}{3} $

$ P(A \cup B) = 1 - \frac{1}{3} = \frac{2}{3} $

Final Answer Calculation

To round the probability to two decimal places, we convert the fraction to a decimal:

$ \frac{2}{3} \approx 0.6666... $

Rounding to two decimal places gives $0.67$.

The calculated probability $0.67$ falls within the range specified (between 0.66 and 0.67).