In conditional probability, a phrase limiting which items can be selected changes the group used for the probability. The group selected from sets the denominator. Find that group's total in the table, then count the requested outcome inside it. If a count is unknown, write the probability equation and use a Desmos regression to find the value that makes both sides equal. Don't use the whole table's total.
Hints
- Hint 1
A condition limits which items could be picked. Because the selection is from failed widgets, look only at the failed row. What is that row's total when you add its red and blue counts?
- Hint 2
A favorable outcome is an item that matches the requested color. Blue failures are the favorable items here. Put their count over the total number of failures, then set that fraction equal to the given probability.
Step-by-step
Use the failed row, then solve in Desmos
Step 1Find the group that can be selected
The widget is selected from failed widgets, so only the Failed inspection row can be used. That row has red widgets and blue widgets, making its total . The and passed widgets are outside this group.
- Step 2
Write the probability equation
Probability is the favorable count, the number matching what you want, divided by the total number that could be picked. The blue failures occupy the cell of the failed row, so match their fraction to the given probability:
- Step 3
Find the missing count
To use a Desmos regression, replace with and with ; Desmos then finds a value that makes the sides match rather than graphing an equation in . Type . Desmos shows under PARAMETERS, so 20 blue widgets failed inspection. Grid in 20.