Question

The following questions are from ARTIST (reproduced with permission) If Tahnee flips a coin 10 times, and records the results (Heads or Tails), which outcome below is more likely to occur, A or B? Explain your choice. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { Throw Number } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{1 0} \\ \hline \mathbf{A} & \mathrm{H} & \mathrm{T} & \mathrm{T} & \mathrm{H} & \mathrm{T} & \mathrm{H} & \mathrm{H} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\\ \hline \mathbf{B} & \mathrm{H} & \mathrm{T} & \mathrm{H} & \mathrm{T} & \mathrm{H} & \mathrm{T} & \mathrm{H} & \mathrm{T} & \mathrm{H} & \mathrm{T} \\\ \hline \end{array} $$

Short answer

Both outcomes are equally likely.

Step-by-step solution

Understand Equal Probability

Each coin flip is independent, so each flip of the coin has two possible outcomes: Heads (H) or Tails (T). The probability of getting Heads is 0.5, and the probability of getting Tails is also 0.5.

Recognizing Pattern Complexity

Outcome A has a non-repeating pattern over the 10 flips: H, T, T, H, T, H, H, T, T, T. Outcome B has a repeating pattern of H, T over the 10 flips. This might seem distinct but doesn't affect likelihood.

Probability Calculation of Each Pattern

Calculate the probability of each specific sequence. Since each flip is independent, the probability of any sequence of 10 coin flips occurring is \( (0.5)^{10} \). This applies to both sequence A and B.

Final Decision

Since both outcomes (A and B) have the same probability \( (0.5)^{10} \), neither is more likely than the other. Both sequences of flips occur with the same probability.

Key concepts

Independent Events

When discussing probability and outcomes, especially in scenarios like coin flipping, the concept of independent events is crucial. An event is considered independent if its outcome does not affect the outcome of another event.
With coin flipping, each flip is independent of the previous ones. This means the outcome, whether it lands on Heads (H) or Tails (T), does not influence the result of the next flip.
This is an important property and a defining characteristic of truly random processes. In any series of coin flips, such as flipping a coin 10 times, each flip remains an independent event. The probability of getting a Head remains 0.5, and a Tails is also 0.5, regardless of what the previous results were.
Understanding independent events ensures that one can correctly compute probabilities without incorrectly assuming any pattern influence from prior outcomes.

Coin Flipping

Coin flipping is a classic example of probability in action. It's a simple, intuitive activity used frequently to show basic principles of randomness and probability.
Each side of the coin has an equal chance of landing face up.
If we denote Heads by H and Tails by T, then the outcomes of each flip are ***H*** or ***T***.

• A *fair coin* ensures that the chances of H or T are equal.
• During the process, the history of previous flips doesn’t alter future results due to the principle of independent events.

This makes each flip a great example of a purely random event. Learning through coin flipping lays the foundation for understanding more complex real-world systems where probability plays a key role, such as genetics, game design, and risk assessment in financial services.

Probability Calculation

Calculating probability, especially with repetitive events like coin flipping, helps determine the likelihood of various outcomes.
For a single coin flip, the probability of landing on Heads is 0.5, and the probability of landing on Tails is also 0.5.
When we're dealing with multiple flips, the task is to consider all of them collectively.

• The probability of a specific sequence occurring (like H, T, H, T, H) is found by multiplying the probability of each individual flip:

If Tahnee flips a coin 10 times, each flip is independent, and so the probability of any particular sequence occurring is \( (0.5)^{10} \). This calculation is the same for any specific sequence of 10 coin flips, be it alternating or random in appearance. This computation stems from the fundamental principle that the probability of multiple independent events all occurring is the product of their individual probabilities.