Please solve the statistical questions below keenly. Provide clear explanations

We consider three groups of policyholders: A, B and C. We denote by X, the random variable for the number of claims that a randomly chosen policyholder in group A submits during a calendar year. Mp and X denote the corresponding random variables for policyholders in groups B and C. We assume that Xy, \\, and Y have Poisson distributions with parameters ), =0.2. Ag =0.1 and Ac =0.05 depending on the group. Each policyholder belongs to exactly one group and group membership does not change during the lifetime of a policyholder It is assumed that any individual policyholder submits claims during any year independently of claims submitted by other policyholders. An insurance company has a portfolio of policies with 20% of policyholders belonging to group A, 20% belonging to group R and the remaining policyholders belonging to group C. The insurance company chooses a policyholder at random. Determine the probability that this policyholder will submit at least two claims during a year given that he belongs to group A. [2] The insurance company chooses another policyholder at random but does not know to which group he belongs. (ii) Show that the probability this policyholder will submit exactly one claim during a year is approximately 0.0794. [3] (iii) Calculate the probability that this policyholder belongs to group A given that he submitted exactly one claim in the previous year. [2] [Total 7] In a small empirical study 100 male and 100 female workers in a company are asked about their body weight and then classified into the following three categories: not overweight, overweight and obese. The observed numbers of workers in each category are shown in the following table. Not overweight Overweight Obese Total Females 45 32 23 100 Males 33 41 26 100 Total 78 73 49 200 Test the null hypothesis that weight classification is independent of gender, using a 5% significance level. [5]One card is picked from an ordinary pack of 52 playing cards. What is the probability of obtaining: a diamond (ii) an ace (iii) the ace of diamonds (iv) a jack, queen or king