Coin Toss Outcomes and Sample Space

When two coins are simultaneously tossed, we need to determine all the possible results that can occur. This collection of all possible results is known as the sample space.

Let 'H' represent a head and 'T' represent a tail. When tossing two coins, the outcomes for each coin are independent. The possible combinations are:

• The first coin lands on Heads (H) and the second coin lands on Heads (H). This outcome is denoted as HH.
• The first coin lands on Heads (H) and the second coin lands on Tails (T). This outcome is denoted as HT.
• The first coin lands on Tails (T) and the second coin lands on Heads (H). This outcome is denoted as TH.
• The first coin lands on Tails (T) and the second coin lands on Tails (T). This outcome is denoted as TT.

Therefore, the complete sample space (\(S\)) for tossing two coins simultaneously is:

\(S = \{HH, HT, TH, TT\}\)

The total number of possible outcomes in the sample space is \(n(S) = 4\).

Probability of Two Heads Appearing

We are interested in the event where two heads appear simultaneously. Let's call this event \(E\).

From our sample space \(S = \{HH, HT, TH, TT\}\), the outcome that consists of two heads appearing is only \(HH\).

So, the event \(E = \{HH\}\).

The number of favorable outcomes for event \(E\) is \(n(E) = 1\).

Calculating Probability

The probability of an event is calculated using the formula:

\[ P(E) = \frac{\text{Number of favorable outcomes for event E}}{\text{Total number of possible outcomes in the sample space}} = \frac{n(E)}{n(S)} \]

Using the values we found:

• Number of favorable outcomes (\(n(E)\)) = 1 (for HH)
• Total number of possible outcomes (\(n(S)\)) = 4 (for HH, HT, TH, TT)

Now, substitute these values into the probability formula:

\[ P(\text{two heads}) = \frac{1}{4} \]

Thus, the probability of two heads simultaneously appearing when two coins are tossed is \(1/4\).