Question

Shortly after the introduction of the euro coin in Belgium, newspapers around the world published articles claiming the coin is biased. The stories were based on reports that someone has?

Answer 1

The question regards a mathematical experiment to test the fairness of the Belgian euro coin by spinning it 250 times and recording the outcomes. Calculations using the experimental data show that the coin may not be fair. A probability tree helps calculate the likelihood of various outcomes when spinning the coin twice.

Step-by-step explanation:

When the euro coin was introduced in Belgium, concerns about its fairness arose. Two math professors conducted an experiment with their students to investigate this claim. They spun the coin 250 times, resulting in 140 instances of heads (event H) and 110 of tails (event T).

Calculating Probabilities

To calculate the probability of getting heads, P(H), we divide the number of heads by the total spins: P(H) = 140/250 = 0.56. Similarly, for tails, P(T) = 110/250 = 0.44. The professors claimed that the coin is not a fair one, as the probabilities did not equate to the expected 0.5 for a fair coin.

Using a Probability Tree

A probability tree can be used to visualize and calculate the outcome probabilities for two spins of the coin:

• The first branches show the probability of getting heads or tails on the first spin.
• The second set of branches show these probabilities for the second spin.

Each path represents a possible outcome (HH, HT, TH, TT), and the probability of each outcome is the product of the probabilities along that path.

Calculating Specific Outcomes

To find the probability of getting exactly one head in two spins:

1. Identify the paths with exactly one head (HT, TH).
2. Calculate the probability of each path: P(HT) = P(H) × P(T) and P(TH) = P(T) × P(H).
3. Add the probabilities: P(exactly one head) = P(HT) + P(TH).

To discover the probability of getting at least one head:

1. Combine the probabilities of getting one or two heads.
2. Add the probabilities of HT, TH, and HH.

These calculations show the usefulness of a probability tree in understanding complex probabilistic scenarios.

Answer 2

The stories were based on reports that someone had spun the coin 250 times and gotten 140 heads that's 56% heads.

Explanation

Using normal distribution and finding probability and then deriving hypothesis from the above, it could be ascertained if the euro was truly biased or unfair.