Thursday, 9 May 2013

Common Coincidence

After reading ‘Check, Please!’1 on www.theoddsmustbecrazy.com I wanted to investigate how common is is for the total bill for a group of 4 people to be the same as another group of 4 people in the restaurant. In theory it must occur fairly frequently when the menu is fairly small or has similar prices. First of all, I needed to find a restaurant menu I could use to obtain some reasonable prices in order to investigate this. I chose the USA McDonalds breakfast menu 2, because then I would be working with a smaller range of foods and drinks, but the results would be representative of the situation described in the article. I then programmed a Monte Carlo simulation in order to look at all the random total bills that could be generated by a group of 4 people, assuming that all 4 ate one meal and one drink, and then I programmed in an optional extra, to make the data more representative of real life, so this would mean one person in the group ordered an extra item of food or not. 

I ran the program for 1000 groups of 4 people, looking at the total bill for each group. I then took this data and produced a histogram to better look and the spread and frequency of each total. As expected, the values in the middle (in between the cheapest possible combination and the most expensive possible combination) occurred most frequently, because there were many ways to obtain this value from different combinations of food and drink, but that the extreme values are far less frequent because there are fewer different combinations that could produce this result. So we see a normal distribution, with a characteristic bell curve.

I then investigated how often two random groups would have the same total value by examining my data and producing a frequency table. As the sample size increases the likelihood of recording the same value increases, so for 100 groups 60% of the totals occur at least twice, and for 1000 groups 97% of the totals occur at least twice. We can see this when we look back at the histogram, as the majority of the values have a bar with a height of more than 2. So if you are working in a busy restaurant and serve 100 people in a day, and 60% of those bills are likely to have occurred twice, why do we not notice them?
Total bill ($)
Frequency
Total bill ($)
Frequency
Total bill ($)
Frequency
9.1
1
12.05
10
14.6
17
9.2
1
12.1
6
14.65
13
9.25
2
12.15
6
14.7
7
9.3
1
12.2
10
14.75
13
9.35
1
12.25
7
14.8
15
9.45
1
12.3
4
14.85
13
9.55
2
12.35
7
14.9
6
9.6
1
12.4
9
14.95
10
9.65
1
12.45
6
15
9
9.7
1
12.5
6
15.05
7
9.85
1
12.55
6
15.1
10
9.95
1
12.6
10
15.15
12
10
3
12.65
7
15.2
15
10.05
        1
12.7
10
15.25
10
10.1
4
12.75
12
15.3
9
10.15
2
12.8
15
15.35
9
10.2
1
12.85
14
15.4
11
10.3
1
12.9
12
15.45
7
10.35
1
12.95
9
15.5
2
10.4
3
13
7
15.55
4
10.45
1
13.05
16
15.6
10
10.5
1
13.1
13
15.65
2
10.55
2
13.15
12
15.7
4
10.6
2
13.2
15
15.75
6
10.65
1
13.25
14
15.8
3
10.7
3
13.3
14
15.85
4
10.75
5
13.35
12
15.9
5
10.8
2
13.4
10
15.95
2
10.85
5
13.45
9
16
6
10.9
4
13.5
8
16.05
5
10.95
5
13.55
7
16.1
5
11
2
13.6
4
16.15
3
11.1
5
13.65
12
16.2
7
11.15
2
13.7
16
16.25
2
11.2
3
13.75
15
16.35
5
11.25
5
13.8
14
16.4
4
11.3
4
13.85
9
16.45
2
11.35
2
13.9
19
16.5
2
11.4
1
13.95
10
16.55
2
11.45
5
14
13
16.6
2
11.5
5
14.05
10
16.65
1
11.55
1
14.1
13
16.7
1
11.6
5
14.15
10
16.75
1
11.65
6
14.2
16
16.8
2
11.7
9
14.25
15
16.85
2
11.75
8
14.3
16
16.9
1
11.8
5
14.35
10
16.95
1
11.85
5
14.4
13
17.1
2
11.9
7
14.45
12
17.25
1
11.95
4
14.5
11
17.35
1
12
7
14.55
13
18.35
1

I went on to look at the list of random totals in the order they were generated in and I sorted for those whose value is exactly the same as the next total, to see how frequently you could expect two bills, one after another to occur, within a sample of 1000 random groups. Only 0.6% of totals have the same value occur twice in a row. This could explain why the chance of the same total occurring is very common, but because it doesn’t occur immediately after the first appearance of that total we do not notice it, and same totals occurring together are very infrequent. This would explain what occurred in the article and gives a realistic probability of how often this would occur in restaurant situations.
E Markham (2013). Common Coincidence Blogspot

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