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A tree diagram is a visual tool that helps illustrate all possible outcomes of a sequence of events, like coin flips. It starts with a single point, representing the initial state before any event has occurred. From there, branches are drawn to represent the possible outcomes of each subsequent event.

For example, with a single coin flip, you have two outcomes: heads (H) or tails (T). If you were to flip a coin three times, the tree diagram would start with one branch branching out into two for the first flip (H or T), and then each of those branches would also split into two more branches for the second flip, and so forth. By the third flip, you end up with eight distinct outcomes.

Using a tree diagram for our exercise, we were able to visualize all the possible outcomes when flipping a coin three times: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.

Probability calculation is the process of determining the likelihood of a particular event occurring. This is done by dividing the number of ways a specific event can occur by the total number of possible outcomes.

In the context of our exercise, we are interested in the event 'exactly one head occurred' during three coin flips. To find its probability, we identify the number of outcomes that match our event from the tree diagram and divide that by the total number of outcomes. With the tree diagram, we can see there are three branches (HTH, THH, HTT) that represent the event out of a total of eight possible outcomes.

The formula for probability, in this case, is quite straightforward: P(Event) = \( \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}} \). For 'exactly one head', P = \( \frac{3}{8} \), as there are three favourable outcomes out of eight total possibilities.

Combinatorics is the branch of mathematics that deals with counting, combining, and arranging sets of elements. In probability, combinatorics helps us count the number of ways events can occur without having to list all possibilities, which becomes especially useful for larger sets of data.

Considering our coin flip problem, combinatorics allows us to calculate the number of ways we can get 'exactly one head' out of three flips without explicitly drawing a tree diagram. We use combinations to determine the number of ways to choose 1 head (H) out of 3 flips. The formula for combinations is \( C(n, k) = \frac{n!}{k!(n-k)!} \), where 'n' is the total number of items, 'k' is the number of items to choose, and '!' denotes factorial (the product of an integer and all the integers below it).

For our exercise, we calculate \( C(3, 1) \), which simplifies to \( \frac{3!}{1!(3-1)!} = 3 \). This calculation confirms that there are indeed three ways to have a single head appear in three coin flips, aligning with our tree diagram's visual representation.