Calculate the average number of rolls of one dice needed to roll two sixes in a row. Instructions on how to proceed: a) The assignment means that we will roll the dice until two sixes fall in a row....

Extracted text: Calculate the average number of rolls of one dice needed to roll two sixes in a row. Instructions on how to proceed: a) The assignment means that we will roll the dice until two sixes fall in a row. The number of throws we've made is the number of throws needed to roll two sixes in a row. Your task is to calculate the average number of rolls of one dice needed to roll two sixes in a row. We get the average number by repeating the given experiment an infinite number of times and calculating the average of the achieved results b) Let's mark a; the probability that in i one consecutive roll does not fall twice 6 in a row and six fall in the last roll. Let bi denote the probability that one dice will not fall in even consecutive rolls six in a row and six in the last roll will not fall. [still b) but maybe better translated: Let us also mark the probability that he will not fall by one dice in even consecutive throws twice in a row and six in the last roll. Let's mark the probability that one dice will not fall in one successive throw six in a row and six in the last roll] c) Probabilities a; , b; form sequences {a;}=o and {b;}=0- Their first members are a, = 0 and bo = 1. Assemble a set of recurring relationships for sequences {a;}%=0 and {b;}%-0 and find the creating functions ax) and b(x) for this sequences. 100


Extracted text: d) Let's choose p; probability of, that in i consecutive throws with one dice will fall exactly two sixes in a row in the last two throws. e) Probabilities pi form probability {p;}o. which first elements are po= 0 and p1 = 0. k=0? Find a creating function p(x) for probability {Pi}o; Šk=0• f) The average number of dice roller throws required to throw two sixes in a row is calculated as: P = E, i Pi (Consider this formula as fiven fact, you don't have to prove it) g) Use formula P=Sim to calculate the required average number of throws needed to i=0 throw two sixes in a row with one dice.