Question

1.A biased die is initially thrown 25 times and the number of 1's seen face-up is seven. If the die is then thrown an additional 12 times, find the probability that a 1 will be seen exactly three times. Round your final answer to four (4) decimal places.

2.A biased die is initially thrown 25 times and the number of 1's seen face-up is seven. If the die is then tossed an additional 12 times, find the probability that a 1 will be seen in at least half of those additional tosses. Round your final answer to four (4) decimal places.

3.A biased die is initially thrown 25 times and the number of 1's seen face-up is seven. If the die is then thrown an additional 12 times, find the expected (mean) number of 1's tossed. Round your final answer to two (2) decimal places.

4.A biased die is initially thrown 25 times and the number of 1's seen face-up is seven. If the die is then thrown an additional 12 times, find the standard deviation of the number of 1's. Round your final answer to four (4) decimal places.

Answer 1

Let X denote the number of times the number of 1's seen face-up. Then using the information given we know that X follows a Binomial distribution with n=12 and p=7/25=0.28. thus the pdf of X is given by,

a) The probability that a 1 will be seen exactly three times = P(X=3) = 0.2511

b) The probability that a 1 will be seen in at least half of those additional tosses = P(X6) = 0.0887

c) E(X) = np = 3.36

d) Standard deviation is =1.5554