Final answer:
Using the binomial probability formula, the probabilities when selecting 5 video games are 0.000377 for all being rated Mature, 0.484 for none being Mature, 0.516 for at least one being Mature, and 0.013 for exactly three being Mature.
Step-by-step explanation:
To find the probability of various outcomes when choosing 5 video games at random, given that 13.2% are rated Mature, we can use the binomial probability formula:
P(X = k) = C(n, k) × (p^k) × ((1-p)^(n-k))
where C(n, k) is the binomial coefficient, n is the total number of trials, k is the number of successful trials, and p is the probability of success on a single trial.
- a. Probability that all 5 games are rated Mature: This is P(X=5), so (0.132^5)×5!/(5!0!) = 0.132^5 = 0.000377.
- b. Probability that none of the 5 games are rated Mature: This is P(X=0), so (0.868^5)×5!/(5!0!) = 0.868^5 = 0.484.
- c. Probability that at least one game is rated Mature: This is 1 - P(X=0), which is 1 - 0.868^5 = 0.516.
- d. Probability that exactly 3 games are rated Mature: This is P(X=3), so C(5, 3) × (0.132^3) × (0.868^2) = 10 × 0.132^3 × 0.868^2 = 0.013.
Thus, the probabilities are 0.377 for all games Mature, 0.484 for no games Mature, 0.516 for at least one game Mature, and 0.013 for exactly three games Mature, all rounded to three decimal places.