Simple Probability:Compute these probabilities: a. The probability of rolling the number 6 using one die. b. The probability of rolling the number 6 twice in 2 rolls using only one die. c. If the probability of seeing a male house finch at a birdfeeder is .45 and the probability of seeing a female house finch is .35; what is the probability of seeing, first a male house finch, then a female house finch, and finally another male house finch? d. There are only 2 type of Pokemon: rare and common. If the probability of finding a rare Pokemon at a Pokestop is .05, what is the probability that the first 8 Pokemon that you find are common, and then you find one rare Pokemon at a Pokestop that has a lure attached to it. Assume that you will find at least 9 Pokemon in the 30 minutes that the lure is attached.1. Joint Probability:An apple juice bottling company maintains records concerning the number of unacceptable bottles of juice obtained from the filling and capping machines. Based on past data, the probability that a bottle came from machine I and was nonconforming is 0.05 and the probability that a bottle came from machine II and was nonconforming is 0.075. These probabilities represent the probability of one bottle out of the total sample having the specified characteristics. Half the bottles are filled on machine I and the other half are filled on machine II.Compute: a. If a filled bottle is selected at random, what is the probability that it is a nonconforming bottle? b. If a filled bottle is selected at random, what is the probability that it was filled on machine II? c. If a filled bottle is selected at random, what is the probability that it was filled on machine I and is a conforming bottle?2. Lottery Probability:Suppose that there is a lottery where you select 3 different numbers between 0 and 9. A digit cannot be selected twice so you can select 123 but not 112.a. How many different three digit numbers are possible? b. If you buy one ticket, what is the likelihood you will win? c. If you buy 3 tickets, what is the likelihood you will lose?3. Probability using the z test:According to Investment Digest (Diversification and the Risk/Reward Relationship, Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 15.4%, and the standard deviation of the annual return was 21.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 7.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously.a. What is the probability that the stock returns are greater than 0%? b. What is the probability that the stock returns are less than 20%?