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Understanding the probabilities in a coin toss is fundamental to grasping more complex probability concepts. When we flip a fair coin – which means that the coin has an equal chance of landing on either side – we are dealing with one of the most basic examples of a random event in probability theory.

For a single coin toss, there are two possible outcomes: heads (H) or tails (T). Since the coin is fair, the probability of it landing on heads is the same as that of tails, which is exactly half, or mathematically expressed as \( \frac{1}{2} \). This probability does not change no matter how many times you flip the coin, which means each flip is an independent event.

• First Flip: \( \frac{1}{2} \) chance of heads
• Second Flip: \( \frac{1}{2} \) chance of heads, independent of the first flip's result
• Third Flip: \( \frac{1}{2} \) chance of heads, independent of the previous flips

This independence is crucial when calculating the probabilities over multiple flips, as subsequent results do not affect the likelihood of heads in any single toss.

A probability tree diagram is an excellent visual tool that helps us see all the possible outcomes of an event and calculate their probabilities. For multiple coin tosses, a tree diagram branches out for each possible outcome, becoming more complex with each consecutive flip.

In our example, where the coin is tossed until a head appears, and we are looking at a maximum of three tosses, the tree diagram starts with a single branch, which then splits into two branches for the second toss, and each of those branches into another two for the third toss, each representing heads or tails. The further along the tree you go, the more branches you encounter, which mirror the increasing number of possible sequences of heads and tails.

Visualizing Outcomes

For each toss, the two possible outcomes (heads or tails) create a fork in the tree. This results in a comprehensive overview of outcomes, where each path from the root to an end node represents a sequence of tosses, along with the accompanying probabilities of that sequence occurring.

Calculating the combined probability of one or more events occurring is frequently done using a probability tree diagram. Each path through the tree represents a unique sequence of events with its own probability, calculated by multiplying the probabilities of each step along the path.

For a head to appear within three tosses, you consider all the paths that result in a head. This includes a head on the first toss, a tail on the first followed by a head on the second, and tails on the first two followed by a head on the third. You find the probabilities for these individual paths by multiplying the probabilities at each branch along the path:

• 1st Toss (heads): \( \frac{1}{2} \)
• 1st Toss (tails) and 2nd Toss (heads): \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)
• 1st and 2nd Toss (tails) and 3rd Toss (heads): \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \)

Finally, to get the total probability of a head appearing at least once in those three tosses, you sum the individual probabilities, which equals to \( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{7}{8} \). This calculation is based on the law of total probability, which allows us to consider all the distinct ways the desired outcome can occur.