Final answer:
The probability of exactly one success is approximately 0.1317, and the probability of less than three successes is approximately 0.4395 when a die is rolled four times, with success being a roll greater than 2.
Step-by-step explanation:
Probability of Success on Die Rolls
When a die is thrown 4 times, and getting a number greater than 2 is considered a success, we can calculate the following probabilities:
(i) The probability of exactly one success can be determined by considering the success as rolling a 3, 4, 5, or 6, which has a probability of 4/6 (since there are four successful outcomes out of six total possible outcomes). The probability of failure (rolling a 1 or 2) is therefore 2/6.
Using the binomial probability formula, the probability of exactly one success in four trials (k=1 success, n=4 trials) is given by:
P(X=k) = (n choose k) p^k (1-p)^(n-k)
We find P(X=1) with p=4/6, q=2/6, n=4:
P(X=1) = (4 choose 1) (4/6)^1 (2/6)^(4-1) = 4 * (4/6) * (2/6)^3 = 0.1317 (rounded to four decimal places).
(ii) The probability of less than 3 successes includes the events of 0, 1, or 2 successes. We calculate each of these probabilities and add them together.
The sum of probabilities for 0, 1, or 2 successes is:
P(X<3) = P(X=0) + P(X=1) + P(X=2)
Calculating these individual probabilities with the binomial formula (and rounding to four decimal places) we get:
P(X=0) + P(X=1) + P(X=2) = 0.0444 + 0.1317 + 0.2634 = 0.4395
Therefore, the probability of getting less than 3 successes is 0.4395.