Final answer:
Marshall's number of spins until winning on a 4-segment prize wheel represents a geometric setting, as it involves binary outcomes, consistent probability, and independent trials. As such, X is a geometric random variable and the probability of winning on a specific spin can be calculated using the geometric probability formula.
Step-by-step explanation:
The scenario described involves Marshall spinning a prize wheel where there are 4 segments, one of which is a winner. This sets up a geometric distribution, as we are looking at the variable X, which represents the number of spins until Marshall wins a prize. To be considered a geometric setting, certain conditions must be met:
- There is a sequence of trials, where each trial only has two possible outcomes (in our case, win or lose), making the outcomes binary.
- The probability of success (winning) is constant in each trial.
- The trials are independent of one another, meaning the outcome of one spin does not affect the outcome of another.
In Marshall's case, since the wheel is divided into equal segments, the probability of winning (p) is 1/4 (since there is one winning segment) and the probability of losing (q) is 3/4. This probability remains consistent across all spins, satisfying the constant probability requirement. After a spin, the outcome does not influence the next spin, so the trials are indeed independent. Given these conditions, we can confirm that X does indeed represent a geometric setting, with Marshall's success being defined as winning the prize.
As an example, to find the probability that Marshall wins on the fifth spin, we would calculate P(X = 5). Since we're dealing with a geometric random variable, the probability that the first win occurs on the fifth spin would utilize the geometric probability formula:
P(X = k) = (1-p)^(k-1) × p
,
where k represents the number of spins until the first success (win).