help solving it

Question

The following table represents food purchase amounts and whether the customer used cash or a credit/debit card. One customer is chosen at random. Give your answer as a decimal out to at least 4 places.

	Less than $10	$10-$49	$50 or More	Total
Cash Purchase	18	20	9	47
Card Purchase	6	8	14	28
Total	24	28	23	75

a) Find the probability that the customer's purchasing method was a cash purchase or the customer spent $50 or more. 

b) Find the probability that the customer's purchasing method was cash purchase and the customer spent $50 or more. 

c) Find the probability that the customer's purchasing method was a cash purchase given they spent $50 or more. 

d) Find the probability that the customer spent less than $50. 

e) What percent of cash purchases were for $50 or More. Round percent to two decimal places. %

Answer 1

Hi Lakotah,

This problem is similar to the one you posted last night--it involves conditional probability, Bayes' Theorem, and the Addition and Multiplication Rules.

a. Use the Addition Rule:

P(A or B)= P(A) + P(B) - P(A and B)

A= Cash

B= Spent $50

P(A)= 47/75

P(B)= 23/75

P(A and B)= 9/75 (Look at row 1 , column 3)

Substitute these values into Addition Rule above, and you will have probability of cash or >$50 purchase.

b. This can be taken directly from row 1, column 3. Corresponding probability is:

9/75= 0.1200

c. Use Bayes' Theorem:

P(A given B)= P(A and B) / P(B)

A=Cash

B=Spent > $50

P(A and B) = 9/75 (Row 1, column 3)

P(B)= 23/75

P(A given B) = (9/75) / (23/75)

Do those calculations and you will have the probability that a patron paid cash given they spent over $50.

d. To find the probability a customer spent less than $50, you need to add the under $10 and $10 to $49 columns together. This is: (24 + 28) = 52

Now, divide that by the total shoppers:

P= 52/75

Do this computation and you will have probability that customer spent less than $50.

e. From table, total number of cash purchases was 47. Of those 47, 9 spent $50 or more. So the probability is:

P= 9/47 = 0.1915

But the question asked for a percent, so we multiply by 100:

P= 19.15%

I hope this helps. I also noticed you have a few problems like this on here. If you like, we can set up an online tutoring session to discuss some additional examples.