Question 33·Hard·Probability and Conditional Probability
A retailer analyzed 1,200 recent online orders and noted whether each order was shipped using express shipping and whether it included paid gift-wrap. The results are summarized in the table below.
| Gift-wrapped | Not gift-wrapped | Total | |
|---|---|---|---|
| Express shipping | 75 | 225 | 300 |
| Standard shipping | 90 | 810 | 900 |
| Total | 165 | 1,035 | 1,200 |
If one of the orders that included paid gift-wrap is selected at random, which of the following is closest to the probability that the order was shipped using express shipping?
For conditional probability questions with a table, first underline the condition ("given" part) in the question and immediately restrict your attention to that row or column as the new total. Then, within that restricted group, identify the count that matches the event of interest ("successes"). Form the fraction (successes)/(new total), and quickly convert to a decimal—using mental math or Desmos—to match the closest answer choice. This avoids common mistakes like dividing by the overall total or accidentally reversing the condition.
Hints
Focus on the correct group of orders
The phrase "If one of the orders that included paid gift-wrap is selected at random" means you should only consider orders that were gift-wrapped. Which number in the table tells you how many orders were gift-wrapped in total?
Identify the successful outcomes
Among the gift-wrapped orders, you want the ones that used express shipping. Which cell in the table gives this count?
Write the probability as a fraction
Once you know (1) the total number of gift-wrapped orders and (2) how many of those used express shipping, write the probability as , then convert that fraction to a decimal and compare with the choices.
Desmos Guide
Use Desmos to compute the conditional probability
In the expression line, type 75/165 and press Enter. Look at the decimal approximation that Desmos displays for this fraction, then choose the answer choice that is numerically closest to that decimal.
Step-by-step Explanation
Recognize this is a conditional probability
The question says: "If one of the orders that included paid gift-wrap is selected at random…".
This means we are only looking at the subset of orders that are gift-wrapped. So we are finding the probability of express shipping given that the order is gift-wrapped.
Find the total number of gift-wrapped orders
From the table, look at the Gift-wrapped column in the Total row.
- Total gift-wrapped orders .
This is the new “total” for our conditional probability.
Find how many of those gift-wrapped orders used express shipping
Still in the Gift-wrapped column, now look at the Express shipping row.
- Number of express, gift-wrapped orders .
These are the “successful” outcomes for our probability.
Compute the conditional probability and match to the choices
The probability that an order used express shipping given that it is gift-wrapped is
Simplify or convert to a decimal:
The value is closest to , which is the correct answer.