Final answer:
The mean number of correct guesses out of 50 trials is 12.5, and the standard deviation is approximately 3.1, assuming the contestants are purely guessing with a one in four chance of being correct.
Step-by-step explanation:
The student is asking about probability and statistics, which is a part of Mathematics. Specifically, they are asking for the mean and standard deviation of the number of correct guesses in a series of 50 trials, assuming the contestants are merely guessing, and there is a one in four chance of selecting the card with the star.
Calculating the Mean
The mean (expected value) is calculated by multiplying the number of trials (n) by the probability of success (p) on a single trial. With four cards, one of which has a star, the probability (p) of guessing correctly is 1/4 (or 0.25). So, for 50 trials (n), the mean number of correct guesses would be:
Mean (μ) = n × p = 50 × 0.25 = 12.5
Calculating the Standard Deviation
The standard deviation (SD) is calculated using the following formula where p is the probability of success, and q is the probability of failure (q = 1 - p):
Standard Deviation (SD) = √(n × p × q)
For our problem: q = 1 - 0.25 = 0.75
SD = √(50 × 0.25 × 0.75) = √(9.375) ≈ 3.1 (rounded to the nearest tenth)
Therefore, assuming the contestants are guessing, we would expect on average 12.5 correct guesses out of 50 trials with a standard deviation of about 3.1.