STA1DCT Data-Based Critical Thinking Assignment

Reference no: EM132507774 , Length: 3 Pages

STA1DCT Data-Based Critical Thinking Assignment - La Trobe University, Australia

Q1. Consider a 45-ball lottery game. In total there are 45 balls numbered 1 through to 45 inclusive. 4 balls are drawn (chosen randomly), one at a time, without replacement (so that a ball cannot be chosen more than once). To win the grand prize, a lottery player must have the same numbers selected as those that are drawn. Order of the numbers is not important so that if a lottery player has chosen the combination 1, 2, 3, 4 and, in order, the numbers 4, 3, 1, 2 are drawn, then the lottery player will win the grand prize (to be shared with other grand prize winners). You can assume that each ball has exactly the same chance of being drawn as each of the others.

(a) Consider a population of size N = 45. How many different random samples of size n = 4 are possible from a population of N = 45. Show workings.

(b) Suppose that you choose the numbers 1, 2, 3 and 4 ahead of the next lottery draw. As a fraction, what is the exact probability that you will win the grand prize in the lottery in the next draw with these numbers?

(c) Continuing on part (b) and again as a fraction, what is the exact probability that you will not win the lottery in the next draw with these numbers? Show workings.

(d) This question is tougher and you need to think carefully about the answer. Recall that the order of the numbers chosen is not important and that each number can only be chosen once. In total, how many combinations are there available that include the numbers 1 and 2but not the numbers 3 or 4? Explain and show workings.

Q2. In a deck of playing cards, there are 52 standard playing cards and 2 Joker cards for a total of 54 cards. Consider an experiment where the deck is properly shuffled and a single card is dealt.

(a) What is the probability that the card dealt is a Joker?

(b) Now suppose that the card is returned to the deck and that the deck is once again properly shuffled. The experiment is repeated another 26 times (i.e. in total 27 trials of the experiment are conducted - including the initial trial in (a) above) where the card that is dealt is returned to the deck after each trial and the deck is properly shuffled again before the next card is dealt. Out of the 27 trials, let X equal the number of Joker cards that are observed.

i. What are all of the possible values that can be observed for X?

ii. What is the expected frequency for the number of Joker cards dealt? Show workings.

iii. Suppose that X = 5 is observed. What is the relative frequency for this many Joker cards dealt?

iv. Now suppose that the experiment is to be repeated but this time with a very large n (number of trials). For a very large n, what should the relative frequency for the number of Joker cards dealt be approximately equal to?

Q3. Consider an experiment that firstly involves rolling a fair six-sided die once and then secondly rolling a fair eight-sided die once. Let X denote the number rolled on the first roll and let Y denote the number rolled on the second roll.

(a) How many possible outcomes are there? That is, how many different observed values for (X, Y) can eventuate from this experiment? Explain how you got your answer.

(b) For the remainder of this question you may assume that the different observed values (outcomes) for (X, Y) are all equally likely to occur. What is the probability associated with each outcome occurring?

(c) Let A denote the event that the sum of the two numbers rolled is in between 9 and 11 both inclusive (i.e. 9 ≤ X + Y ≤ 11). Calculate the probability of event A occurring. Show workings.

(d) What is the probability of A not occurring? Show workings.

(e) Now, let B denote the event that the number rolled on the second roll is exactly 2 digits greater than the number rolled on the first roll (i.e. Y = X + 2). Calculate the probability of event B occurring. Show workings.

(f) Are events A and B mutually exclusive? Explain.

(g) What is the probability that both event A and event B occur at the same time. Show workings.

(h) What is the probability that either event A or event B occurs (this includes the possibility of both occurring). Show workings.

(i) What is the probability that event B does occur but event A does not occur? Show workings.

Reference no: EM132507774

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