Probability, Simulation

1. It's lotto time! I! But instead of justtaking a sequence of numbers like normal, we care about the sum of the numbers in the sequence. You win big money if the sum of the 3 numbers is an even number not less than II]. There are l stickers in a box with numbers II}, 1= 2, 3, 4, 5, I5, 7", E, 9. We take 3 stickers (without putting them back) from the box. \"mat is the probability that the sum of these 3 numbers is an even number not less than 1'0 and that you win the lotto? Solve this analytically (enumeration is not allowed) and experimentally by simulation, in each case printing the value found. _ After giving the lotto a try you decide to go back to Clayton Campus for some more gambling action. They are playing a die tossing game where you win a brandnew home if you roll all sides of the die within T rolls. Your crystal ball can see into the flJtlJIE: and tells you that when you roll the die 7 times you don't see the same number appear twice in a row. You decide to bet your entire rent for the year and try to win the house. You toss a fair sixsided die T times. Consider the following events, A = \"each value appears at least once" and E = \"the outcome is alternate in numbers (i_e_, a specific face of the die cannot be observed to occur directly after it was observed on the previous roll but it may have occurred elsewhere in the sequence of rolls)". 'What is p(Al E)? Are you likely to win the house? Solve this analytically (enumeration is not allowed) and experimentally by simulation, in each case printing the value found