Question 21·Hard·Probability and Conditional Probability
An e-commerce site tracked 1,450 orders last week and recorded two pieces of information about each order:
- whether the order contained at least one clothing item, and
- whether the order was shipped internationally.
The counts for three of the four possible combinations are shown in the two-way table below.
| Shipped internationally | Shipped domestically | Total | |
|---|---|---|---|
| Contained clothing items | 180 | 320 | 500 |
| Did not contain clothing items | ? | 540 | ? |
| Total | ? | ? | 1,450 |
If an order shipped internationally is selected at random, which of the following is closest to the probability that the order did not contain any clothing items?
For two-way table probability questions, first complete the table using row and column totals so every cell and margin is known. Then translate the wording carefully: “given [condition]” means restrict to that row or column, so the denominator is the total in that row or column, and the numerator is the cell that also satisfies the event of interest. Finally, form the fraction and use your calculator to get the decimal, picking the closest answer choice.
Hints
Use the totals to fill in missing column sums
Start by adding the known entries in the "Shipped domestically" column and subtracting from the total 1,450 orders to find how many were shipped internationally.
Find the total number of orders without clothing
You know how many orders had clothing items. How can you use the total number of orders to get the number that did not have clothing items?
Determine the missing cell for 'international and no clothing'
Once you know the total number of orders without clothing and how many of those were domestic, subtract to find how many were international and without clothing.
Apply conditional probability
For the probability of "no clothing" given international, your denominator should be the total number of international orders. What count should go in the numerator?
Desmos Guide
Use Desmos to find the international total
In Desmos, type 1450 - (320 + 540) to compute the total number of international orders. The output is the column total for "Shipped internationally."
Use Desmos to find the number of international orders with no clothing
Next, type 1450 - 500 to find the total number of orders without clothing, then subtract the domestic-without-clothing count by typing [answer_from_previous] - 540 (or simply 950 - 540 if you already know 950). The result is the count of orders that were both international and had no clothing.
Use Desmos to compute the conditional probability
Finally, type 410 / 590 (or the two values you just found for international-no-clothing and total international). The decimal output is the probability that an order had no clothing items given that it was shipped internationally; match this decimal to the closest answer choice.
Step-by-step Explanation
Complete the known row and column totals
Use the totals already given in the table.
- In the "Contained clothing items" row, , which matches the row total, so this row is complete.
- Add the domestic column entries: . This is the total number of domestic orders.
- Since there are 1,450 total orders, the number of international orders is
So the column total for "Shipped internationally" is 590.
Find the total number of orders without clothing
We know there are 500 orders with clothing items. So the number of orders without clothing items is
This 950 is the row total for "Did not contain clothing items."
Find how many orders were both international and had no clothing
Now focus on the "Did not contain clothing items" row:
- Row total (no clothing): 950
- Domestic & no clothing: 540
- International & no clothing: unknown
So the missing cell is
You can also confirm by using the international column: total international is 590, and 180 of those had clothing, so international orders had no clothing.
Set up and compute the conditional probability
We want the probability that an order did not contain clothing items, given that it was shipped internationally.
In a table, "given international" means we restrict to the international column:
- Number of international orders with no clothing: 410 (numerator)
- Total number of international orders: 590 (denominator)
So the conditional probability is
Among the answer choices, this corresponds to choice C) 0.69.