Please show all your work for each parts thank you

3. A neighborhood Christmas raffle has a total of 50 raffle tickets and features 10 big prizes and 15 smaller ones. (The remaining 25 tickets win no prizes.) A family of 4 participates in the raffle and selects 4 tickets. (a) Let X denote the number of prizes won by the family. Compute the probability mass function of X. (1 point) (b) Let Y denote the number of small prizes won by the family. Compute the cumulative distribution function of Y; (c) Compute the expected number of large prizes won by the family. 4. Students arrive at a tutoring center to a Poisson process with a rate of 10 per hour. (a) What is the probability that the second arrival is observed less than 15 minutes after the first arrival? (b) What is the probability that the third arrival is observed less than 10 minutes after the second arrival if it has not yet been observed 5 minutes after the second arrival? (c) What is the probability that more than 20 minutes is required for the third arrival? (d) Suppose a volunteer working at the center has a 6-hour shift. Compute the expected value and the variance for the number of students who would arrive during her shift