A box contains 20 thermometers, 3 of which are defective. One person randomly draws 2 thermometers from the box, one by one, without replacement. The probability in percent (rounded off to two decimal places) that none of these TWO thermometers is defective, is ______ %.
Show Hint
Use the multiplication rule for dependent events: P(1st good) times P(2nd good given 1st good), then convert to a percentage.
Step 1: Switch to a combinations view.
Instead of multiplying draw by draw probabilities, count the ways to pick 2 thermometers out of 20 and the ways to pick 2 good ones out of the 17 good thermometers. Order of drawing does not matter for this count.
Step 2: Count the total ways to choose 2 out of 20.
\[ \binom{20}{2}=\frac{20\times19}{2}=190 \]
Step 3: Count the favourable ways, both from the 17 good thermometers.
\[ \binom{17}{2}=\frac{17\times16}{2}=136 \]
Step 4: Form the probability as favourable over total.
\[ P=\frac{\binom{17}{2}}{\binom{20}{2}}=\frac{136}{190}=\frac{68}{95} \]
Step 5: Convert to a percentage.
\[ \frac{68}{95}\times100\approx71.58\% \]
This matches the sequential draw by draw calculation exactly, since choosing an unordered pair or drawing one after another without replacement give the same probability.
