# STAT20060: A Casino Offers A Variety Of Gambling Opportunities To Its Customers, Including An Extensive Range Of “One-Armed Bandit” Machines: Statistics & Probability Assignment, UCD, Ireland

University |
University College Dublin (UCD) |

Subject |
STAT20060: Statistics & Probability |

**Q1** A casino offers a variety of gambling opportunities to its customers, including an extensive range of “one-armed bandit” machines. These machines sometimes pay out a jackpot and the mean time between such pay-outs for the casino is 3 hours and 14 minutes. If these can reasonably be modeled with a Poisson distribution how likely is it that the number of jackpots in a 10-hour period will be at least 4?

**Q2** The amount paid out in a jackpot by the one-armed bandits can be modeled as being normally distributed with a mean €121 and a standard deviation €48. What is the probability that the next jackpot pay-out will be between €83 and €155 inclusive?

**Q3** A bet based on a card game is available to players whereby a number of cards are removed by the dealer from a deck of 52, leaving 10 hearts, 9 diamonds, and 12 clubs. From these, a total of 7cards are drawn and the player wins if the number of hearts selected is 3. How likely is it that a player will win such a bet?

**Q4** A different card game involves the dealer taking any 7 cards from the pack. The player must blindly select and replace one of these 7 cards a total of 3 times. They win the game if they manage to select any card more than once. How likely is it that a player will win this game?

**Q5** Players can participate in a daily lotto game where they pick 5 numbers from the set 1, 2 ,… 8. The winning selection is determined by drawing balls from a drum – any player whose selection matches the drawn ball is a winner. If 120 players have entered this draw on one particular day what is the probability that there will be at least 4 winners?

**Probability assignment 2022 instructions**

Often when we wish to estimate the probability of an event we can employ constructs such as probability models, counting principles etc. to get an exact answer. If this approach fails an alternative might be to simulate the experiment in question and estimate the probability empirically.

The probability assignment requires that you take five problems in probability and analyze the probability of a particular event resulting from a (statistical) experiment. In particular, you must calculate the exact or true probability (obtained using probability models/counting principles, etc.) , develop a computer simulation of repeated instances of the experiment and estimate the probability empirically. You should carry out an assessment of how good the estimate it by quantifying its precision (as measured by the standard deviation)

and its accuracy (how close the estimate is to the true value). Finally, an error check statistic should be calculated, which you will find useful in verifying that your workings are correct.

The particular questions you will be required to consider can be found on Moodle – simply locate the file with your name / ID. The five will be based around one of a number of possible scenarios (e.g. holiday resort, film festival, etc.). You will likely have the same scenario as some of your classmates but the

parameters, and hence the answers will be different.

Each of the five problems will be tackled by developing a function in R to be called q1() for question 1, q2() for question 2, etc. The five values to be determined for each question should be returned from the function.

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