Question 950·Hard·Math

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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?

A conditional probability question narrows the random pick to a specified group. Restrict the group first, then count within it. In a two-way table, the named column becomes the denominator, and the target cell in that column becomes the numerator. If that cell is blank, fill it using the table totals. Dividing by the grand total instead answers a different question.

Hints

Hint 1
A condition is something already known about the selected order. Since its shipping destination is known, which column contains every order that could still be picked? Use that column’s total, not the total for the whole site.
Hint 2
The missing cell contains orders without clothing that shipped internationally. The $500$ clothing orders and the $540$ domestic orders without clothing are outside that cell. Subtract both groups from all $1,450$ orders.
Hint 3
Add the missing count to the $180$ clothing orders in the international column. A conditional probability puts the target count over the total for that same column.

Step-by-step

Fill the missing cell, then use the international column
Step 1
Choose the group the order comes from
The order is known to have shipped internationally, so only that column can supply the selected order. For this conditional probability, the international column total is the denominator, and the cell for international orders without clothing is the numerator:
$P(\text{no clothing}\mid\text{international})=\frac{\text{international orders without clothing}}{\text{all international orders}}.$
The bar means “given that.” The $1,450$ orders include domestic shipments, so they aren’t the denominator.
Step 2
Find the missing international count
All $500$ clothing orders and all $540$ domestic orders without clothing are outside the missing cell. Those groups don’t overlap, so subtract both from the $1,450$ orders. Type $1450-500-540$ in Desmos; it shows $410$ . That’s the number of international orders without clothing.
Step 3
Find the international column total
The international column contains $180$ orders with clothing and $410$ without clothing, so add them for the column total. Keep the first Desmos line and type $180+410$ ; it shows $590$ . This is the denominator.
Step 4
Divide within the selected group
Type $410/590$ below the other Desmos lines; it shows $0.694915\ldots$ . That’s the share among international orders that had no clothing, so the closest probability is $0.69$ . Choice C.

Question 950·Hard·Math

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