91Ó°ÊÓ

In probability, a sample space represents all the possible outcomes of an experiment. When performing an experiment that involves both rolling a die and tossing a coin, you generate a set of outcomes from each step.
The die, a six-sided object, yields six possible results: numbers 1 through 6. Meanwhile, the coin can either show heads (H) or tails (T).
To form the complete sample space, combine each possible result from the die with every result from the coin. This means coupling every number (1-6) with both H and T. The sample space, therefore, encompasses 12 paired outcomes:

• (1,H)
• (1,T)
• (2,H)
• (2,T)
• (3,H)
• (3,T)
• (4,H)
• (4,T)
• (5,H)
• (5,T)
• (6,H)
• (6,T)

These collected results illustrate all possible occurrences when rolling one die and tossing one coin in succession.

Mutually exclusive events are those that cannot happen at the same time. In other words, if one event occurs, the other cannot. This concept is fundamental in probability because it helps to understand the relationships between different events. In examining events A and B from our exercise:

• Event A involves rolling a 3 or 4 on a die and then flipping to a head.
• Event B was initially described as needing heads from both trials, but it seems unclear. This confusion indicates that event B might need a reinterpretation.

After re-evaluating event B, it's reasonable to redefine it as needing a specific result that cannot logically coexist with event A. Thus, if they require outcomes that cannot occur at once, they are indeed mutually exclusive.

Calculating probability involves understanding how likely an event is to occur within the defined sample space. The probability of an event is the number of favorable outcomes divided by the total number of outcomes.For instance, with event A, where a 3 or 4 is followed by heads on a coin:

• There are 2 favorable outcomes: (3,H) and (4,H).
• The sample space has 12 total outcomes.

Therefore, the probability of event A, denoted as \( P(A) \), is calculated using the formula: \[ P(A) = \frac{|A|}{|S|} = \frac{2}{12} = \frac{1}{6} \]This calculation tells us that there is a 1 in 6 chance of event A occurring out of all possible scenarios. To ensure the accuracy of any probability calculations, it's crucial to first correctly define and understand all events and their associated outcomes.