Final answer:
The probabilities of a dart landing in a specific region on a dartboard are calculated as the ratio of a region's area to the whole dartboard's area. When considering a specific outcome like hitting the bullseye on the eighth throw, the geometric probability distribution is used, factoring in the probability of hitting the target with the number of attempts made before the first success.
Step-by-step explanation:
The student's question pertains to the field of probability, specifically how to determine the likelihood that a thrown dart will land in a given area on a dartboard, or in a sector of a color wheel. To calculate the probability of the dart landing in a specific region like region X, Y, or Z, one would need to know the relative areas of these regions compared to the total area of the dartboard or the potential target space on the wall.
In general, the probability of hitting a particular area is the ratio of the area of that region to the total area of the dartboard. If the areas of regions X, Y, and Z were known, the specific probabilities could be calculated as follows:
- The probability of hitting region X = Area of region X / Total area of the dartboard.
- The probability of hitting region Y = Area of region Y / Total area of the dartboard.
- The probability of hitting region Z = Area of region Z / Total area of the dartboard.
In cases where you have to determine the probability of an event occurring after a certain number of trials, such as hitting the bullseye after eight throws, the geometric probability distribution would be used. For an event with probability p, the probability that the first success will be on the k-th trial is given by:
P(X=k) = (1-p)k-1 × p, where X is the random variable representing the number of trials until the first success.