Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?
Step-by-Step Solution
Key Concept: A fair experiment is one in which all elementary outcomes are equally likely. In a coin toss the sample space consists of two outcomes – Head (H) and Tail (T) – each occurring with probability \(\frac12\). Hence each team has an equal chance of winning the toss, making the method fair.
1. Identify the sample space: For a single toss of a fair coin, the set of elementary outcomes is \(S = \{H, T\}\).
2. Assign probabilities: Since the coin is unbiased, the probability of each outcome is
$$P(H) = P(T) = \frac{1}{2}.$$
3. Interpret the outcomes: Let Team A get the ball if the result is Head and Team B if the result is Tail (or vice‑versa).
4. Compute the probability for each team:
$$P(\text{Team A gets the ball}) = P(H) = \frac{1}{2},$$
$$P(\text{Team B gets the ball}) = P(T) = \frac{1}{2}.$$
5. Conclude fairness: Both teams have exactly the same probability (\(\frac12\)) of being selected. Since no team is favoured, the coin toss is a *fair* method of deciding which team gets the ball.
6. Link to definition of fairness: According to the definition of a fair experiment in statistics, an experiment is fair if all possible outcomes are equally likely. The coin toss satisfies this condition, therefore it is considered fair.
Correct Answer: Because a fair coin has only two equally likely outcomes – Head and Tail – each occurring with probability \(\frac12\). Assigning one outcome to each team gives both teams an equal chance of getting the ball, so the toss is a fair way of deciding.