SAT Probability Question 27 | Free SAT Prep | Aniko

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Question 27·Hard·Probability and Conditional Probability

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A company uses two machines, Machine A and Machine B, to manufacture identical metal pins. Machine A produces 40% of the pins, and Machine B produces the rest. The defect rate for Machine B is 2 percentage points higher than the defect rate for Machine A. Overall, 4.2% of all pins produced in one day are defective.

The company improves only Machine A so that the overall defective rate becomes 3.6%, while Machine B’s defect rate stays the same.

By how many percentage points must Machine A’s defect rate decrease?

For machine-mixture defect problems, treat each defect rate as a conditional probability and use a weighted average: (share from A)(rate of A) + (share from B)(rate of B) = overall rate. If the situation changes, write a second weighted-average equation for the new overall rate, keeping any “unchanged” machine’s rate fixed, then subtract old and new rates in percentage points.

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Solve for the original defect rates first

Let $x$ be Machine A’s defect rate (decimal) and use $0.40x+0.60(x+0.02)=0.042$ to find $x$ .

Machine B stays the same in the second situation

Once you know Machine B’s defect rate, treat it as a fixed number in the new weighted-average equation for the 3.6% overall defect rate.

Percentage points means subtract the percents

After you find Machine A’s new defect rate, subtract (old %) − (new %) to get the decrease in percentage points.

Desmos Guide

Find Machine A’s original defect rate

In Desmos, enter 0.40x + 0.60(x+0.02) - 0.042 and find the value of $x$ where the graph crosses $0$ (the $x$ -intercept). This gives Machine A’s original defect rate as a decimal.

Use the new overall defective rate to find $x_{\text{new}}$

In Desmos, enter 0.40x + 0.60*0.05 - 0.036 and find the value of $x$ where the graph crosses $0$ (the $x$ -intercept). This value is $x_{\text{new}}$ , Machine A’s new defect rate as a decimal.

Convert to percentage points

Subtract the two decimal rates you found (original minus new), then multiply by $100$ to convert the result to percentage points and match it to the answer choices.

Step-by-step Explanation

Find the original defect rates

Let $x$ be Machine A’s defect rate (decimal). Then Machine B’s defect rate is $x+0.02$ .

Using the overall defective rate as a weighted average:

0.40x + 0.60(x+0.02) = 0.042.

x+0.012=0.042 \Rightarrow x=0.03,

meaning Machine A is at $3\%$ and Machine B is at $5\%$ .

Set up the new overall defective rate after improving Machine A

Let $x_{\text{new}}$ be Machine A’s new defect rate (decimal). Machine B stays at $0.05$ .

The new overall defective rate is 3.6%, so

0.40x_{\text{new}} + 0.60(0.05) = 0.036.

Solve for Machine A’s new defect rate

Compute $0.60(0.05)=0.03$ , so

0.40x_{\text{new}} + 0.03 = 0.036 \Rightarrow 0.40x_{\text{new}}=0.006 \Rightarrow x_{\text{new}}=0.015.

So Machine A’s new defect rate is $1.5\%$ .

Compute the decrease in percentage points

Machine A decreases from $3\%$ to $1.5\%$ , a decrease of $3\%-1.5\%=1.5\%$ (percentage points).

Therefore, the correct answer is $1.5\%$ (choice B).

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation