Springdale Shopping Survey

Probability theory is an essential statistical tool for analyzing data. Probability measures the likelihood that a random event will occur (Soong, 2004). Probability is especially important because it predicts the frequency of specific events or a combination of events (Grinstead & Snell, 2012). The prediction power of probability helps to interpret complex situations for decision making. 

The probability of an event occurring is equal to the ratio of the frequency of that event to the sum of all frequencies. All probabilities fall in the range between  p  (Soong, 2004). Probability of zero means that an event will not occur while a probability of one implies that we are sure that an event will occur (Soong, 2004). Therefore, higher probability means that an event has higher chances of occurring. The summed probability of mutually exclusive events is one. Mutually exclusive events refer to events that cannot co-occur, meaning that the probability of mutually exclusive events coinciding is zero (Grinstead & Snell, 2012). Similarly, the probability of mutually exclusive events occurring is equal to the sum of the individual probabilities. Expressed mathematically: 

Probability = Number of favorable events/ Sum of all events. 

The probability of two mutually exclusive events A and B occurring is: 

P (A or B) = P (A) + P (B) (Grinstead & Snell, 2012). 

The probability of mutually exclusive events A and B occurring together is: 

P (A and B) = 0 (Grinstead & Snell, 2012). 

Table 1 : Probabilities of spending at Springdale 

Springdale
		Frequency	Probabilities	Cumulative Probabilities
$200 or more	1	5	5/150 = 0.03333	0.03333
$150–under $200	2	12	12/150 = 0.08000	0.11333
$100–under $150	3	3	3/150 = 0.02000	0.13333
$ 50–under $100	4	20	20/150 = 0.13333	0.26667
$ 25–under $50	5	29	29/150 = 0.19333	0.46000
$ 15–under $25	6	38	38/150 = 0.25333	0.71333
less than $15	7	43	43/150 = 0.28667	1.00000
Total		150	1.00000

The probabilities of respondents spending different categories of amounts are mutually exclusive, hence the cumulative total of one (Table 1). The probability for each category is obtained by dividing the individual frequencies by the sum of all frequencies. The probability that a respondent spends at least $15 during a trip to Springdale is the sum of probabilities of category $ 15–under $25 and above. 

P (a respondent visiting Springdale would spend at least $15) = P ( $ 15–under $25 ) + P ( $ 25–under $50 ) + P ( $ 50–under $100 ) + P ( $100–under $150 ) + P ( $150–under $200 ) + P ( $200 or more) 

P (a respondent would spend at least $15) = 0.71333 (Table 1). 

Table 2 : Probabilities of spending at Down Town 

Downtown
		Frequency	Probabilities (Relative Frequencies)	Cumulative Probabilities
$200 or more	1	13	13/150 = 0.08667	0.08667
$150–under $200	2	6	6/150 = 0.04000	0.12667
$100–under $150	3	11	11/150 = 0.07333	0.20000
$ 50–under $100	4	6	6/150 = 0.04000	0.24000
$ 25–under $50	5	16	16/150 = 0.10667	0.34667
$ 15–under $25	6	32	32/150 = 0.21333	0.56000
less than $15	7	66	66/150 = 0.44000	1.00000
Total		150	1.00000

P (a respondent visiting Downtown would spend at least $15) = P ( $ 15–under $25 ) + P ( $ 25–under $50 ) + P ( $ 50–under $100 ) + P ( $100–under $150 ) + P ( $150–under $200 ) + P ( $200 or more) 

P (a respondent visiting Downtown would spend at least $15) = 0.56000 (Table 2). 

Table 3 : Probabilities of spending at West Mall 

West Mall
		Frequency	Probabilities (Relative Frequencies)	Cumulative Probabilities
$200 or more	1	5	5/150 = 0.03333	0.03333
$150–under $200	2	12	12/150 = 0.08000	0.11333
$100–under $150	3	6	6/150 = 0.04000	0.15333
$ 50–under $100	4	11	11/150 = 0.07333	0.22667
$ 25–under $50	5	17	17/150 = 0.11333	0.34000
$ 15–under $25	6	29	29/150 = 0.19333	0.53333
less than $15	7	70	70/150 = 0.46667	1.00000
Total		150	1.00000

P (a respondent visiting West Mall would spend at least $15) = P ( $ 15–under $25 ) + P ( $ 25–under $50 ) + P ( $ 50–under $100 ) + P ( $100–under $150 ) + P ( $150–under $200 ) + P ( $200 or more) 

P (a respondent visiting West Mall would spend at least $15) = 0.53333 (Table 3). 

Table 4 : Probabilities for Areas Highest Quality of Goods 

	The highest quality of Goods	Frequency	Probabilities
Springdale Mall	1	79	79/150 =0.52667
Downtown	2	38	38/150 =0.25333
West Mall	3	11	11/150 = 0.07333
No Opinion	4	22	22/150 =0.14667
Total		150	1.00000

The probability that a specific shopping area has the highest-quality goods is the frequency of the particular shopping area divided by the sum of frequencies (Table 4). The brand with the highest probability is perceived as providing the highest quality products and has the highest capacity to attract customers. As shown in (Table 4), Springdale has the highest probability, followed by Downtown, while West Mall has the least. Arranging the brands in descending order, Springdale is the strongest in quality, followed by Downtown, and West Mall is the weakest. 

Table 5 : Probabilities for Gender 

RESPGEND
Gender	RESPGEND	Frequency	Probabilities	Cumulative Probabilities
Male	1	64	0.42667	0.42667
Female	2	86	0.57333	1.00000
Total		150

We use independent probabilities to analyze unrelated events. Two events A and B occurring together are said to be independent if the probability of event A does not affect the probability of event B (Soong, 2004). Gender of a respondent and spending at least $15 at either of the shopping areas are independent. The probability of two independent events occurring together is equal to the product of the individual probabilities. Mathematically, this expressed as: 

P (A and B) = P (A) * P (B) if A and B are independent events (Soong, 2004) 

Table 6 : Probabilities for Spending At Least $15 Given Specific Gender 

	P ( a respondent will spend at least $15)	P (a respondent is Male)	P( a respondent is Female)	P (spends at least $15 and is female) = P( at least $15) * P (female)	P(spends at least $15/male) = P( at least $15) * P (male)
Springdale	0.71333	0.42667	0.57333	0.40898	0.30435
Downtown	0.56000	0.42667	0.57333	0.32107	0.23893
West Mall	0.53333	0.42667	0.57333	0.30578	0.22756

The probabilities of spending at least $15 and being a male or female are calculated by multiplying the individual probabilities (Table 5). Female respondents are more likely to spend at least $15 during their visit to Springdale, Down Town, and West Mall compared to the male respondents. This is because the probability that a respondent is female is higher. Female respondents are more likely to spend at least $15 while shopping at Springdale, followed by Down Town, and are least likely to spend at least $15 at West Mall. Similarly, male respondents are more likely to spend at least $15 while shopping at Springdale, followed by Down Town and are least likely to spend at least $15 at West Mall. 

In conclusion, respondents are more likely to spend at least $15 during their visit to Springdale, Downtown, and West Mall, respectively. Respondents consider Springdale as having the highest quality followed by Downtown and West Mall, respectively. Females are more likely to spend at least $15 at any of the shopping areas. 

References 

Grinstead, C. M., & Snell, J. L. (2012). Introduction to probability. American Mathematical Soc. 

Soong, T. T. (2004). Fundamentals of probability and statistics for engineers. John Wiley & Sons.