1. Probability Theory of Bingo

2. Overview Monte Carlo Method The probability distribution of bingo game How many draws do we need until all gifts aregiven away? participants 80 gifts 20 3. MONTE CARLO METHOD 4. Monte CarloMonte CarloMonte Carlo Casino 5. Monte Carlo Method Monte Carlo Method are a class ofcomputational algorithms that rely onrepeated random sampling to computetheir results.num=10000000;count=0;for i=1:numLets name Monte if 1>rand^2+rand^2 Carlo Method, sincecount=count+1;this algorithm uses end random numbers.endpi = 4*count/numvon Neumann 6. Monte Carlo Method for bingo game Alignment of numbers and order of balls are random. So, I used Monte Carlo Method. 7. THE PROBABILITY DISTRIBUTION OFBINGO GAME 8. The probability distribution of bingo game 9. HOW MANY DRAWS DO WE NEEDUNTIL ALL GIFTS ARE GIVEN AWAY? 10. exWhen 80 participantsMean number of draws until all gifts are given away Mean 95% width 95% CINum of gifts 11. 95%confidence interval [min Max] of number of drawsgifts 5101520 25 30 354045participants min Max min Max min Max min Max min Max min Max min Max min Max min Max 100 19 28 24 31 27 34 30 36 32 37 33 39 35 40 37 42 38 43 95 19 28 24 32 28 34 30 36 32 38 34 40 35 41 37 42 38 44 90 20 29 25 32 28 35 30 37 33 38 34 40 36 42 38 43 39 44 85 20 29 25 33 28 35 31 37 33 39 35 41 37 42 38 44 40 45 80 20 29 25 33 29 36 31 38 33 40 35 41 37 43 39 45 41 46 75 20 30 26 34 29 36 32 38 34 40 36 42 38 44 40 45 41 47 70 21 30 26 34 30 37 32 39 35 41 37 43 39 45 41 46 42 48 65 21 31 26 35 30 37 33 40 35 42 38 44 40 46 42 48 44 50 60 21 31 27 35 31 38 34 41 36 43 38 45 41 47 43 49 45 51 55 22 32 28 36 31 39 34 42 37 44 39 46 42 48 44 51 47 54 50 22 33 28 37 32 40 35 43 38 45 41 48 43 50 46 53 49 57 45 23 33 29 38 33 41 36 44 39 47 42 49 45 53 49 57 55 66 40 23 34 30 39 34 42 38 46 41 49 44 52 48 56 54 66 35 24 35 31 40 35 44 39 47 43 51 47 56 54 66 30 25 37 32 42 37 46 41 50 46 55 53 66 25 26 38 33 44 39 49 44 54 52 65 20 27 40 36 47 42 53 51 65 15 29 43 39 51 49 65 10 32 48 47 64 When (people, gifts) = (80,20) , look red square.5 41 63 Number of draws is between 31 and 38. 12. 5055 6065707580859095100min Max min Max min Max min Max min Max min Max min Max min Max min Max min Max min Max 39 44 41 4642 47 43 48 44 49 46 51 47 52 49 54 51 56 53 59 58 67 40 45 41 4643 48 44 49 45 50 47 52 49 54 51 56 53 59 58 67 41 46 42 4743 49 45 50 47 52 48 54 50 56 53 59 57 67 41 47 43 4844 50 46 51 48 53 50 56 52 59 57 67 42 48 44 4946 51 47 53 49 55 52 59 57 67 43 49 45 5147 53 49 55 52 58 57 67 44 50 46 5249 55 51 58 57 67 46 52 48 5451 58 56 67 48 54 51 5856 67 50 57 56 66 55 66 When you know about number of people and gifts, if you look this table, you can estimate number of draws. 13. Conclusion Monte Carlo Methods algorithms that use many random numbers The probability distribution of bingo game resembles normal distribution mean is about 40 How many draws do we need until all gifts aregiven away? Answered by using my original table