Drawing Coins Without Replacement

What This Problem Teaches

• How "without replacement" changes probability calculations between draws
• The multiplication rule for sequential events (conditional probability)
• Why the total number of outcomes decreases after the first draw
• Setting up organized probability tables to track changing conditions
• Converting probability fractions to percentages for real-world interpretation

Let's Draw It

Solution: Method 1 — Sequential Probability

When drawing without replacement, we treat this as a sequence of events where each draw affects the next. We need a nickel first and a dime second, so we multiply their probabilities.

Step 1 — Count the total coins

First, let's see what we're working with:

Step 2 — Find the probability of drawing a nickel first

On the first draw, we have 10 nickels out of 19 total coins:

Step 3 — Adjust for the second draw

Here's the key insight: after removing one nickel, we have fewer coins total, but the number of dimes stays the same. The bag now contains:

Step 4 — Find the probability of drawing a dime second

On the second draw, we have 4 dimes out of the remaining 18 coins:

Step 5 — Multiply for the combined probability

Since we need both events to happen in sequence, we multiply:

Step 6 — Simplify the fraction

We can reduce this fraction by dividing both numerator and denominator by their greatest common factor of 2:

Solution: Method 2 — Organized Counting

Instead of thinking about conditional probability, we can count all possible ordered pairs and find what fraction are favorable.

Step 1 — Count all possible ordered pairs

For the first coin, we have 19 choices. For the second coin, we have 18 choices (since we don't replace the first). So there are:

Step 2 — Count favorable outcomes

A favorable outcome is picking any of the 10 nickels first, then any of the 4 dimes second:

Step 3 — Calculate the probability

Probability is favorable outcomes over total outcomes:

This gives us the same answer as Method 1, confirming our result.

Verification

Let's check our answer makes sense by testing the boundary conditions and using an alternative approach.

Reasonableness check: We're looking for a specific sequence out of many possibilities, so the probability should be fairly small. 11.7% seems reasonable.

Alternative calculation: We can verify by computing the probability as a decimal directly:

Fraction verification: Let's confirm our simplification is correct:

All checks confirm our answer of 20/171.

Watch Out For These

The Pattern Behind This

This problem demonstrates the fundamental rule for sequential events without replacement:

This pattern extends to any number of draws. For three draws without replacement, you'd multiply three fractions with denominators of n, n-1, and n-2.

The distinction between "with replacement" and "without replacement" is crucial in probability. Without replacement makes each subsequent draw dependent on what happened before, while with replacement keeps each draw independent.

How to Spot This Problem Type

Look for these key phrases that signal a sequential probability without replacement problem:

• "Without replacement" or "don't put it back" — the classic signal
• "Draw two [items]" or "select two [items]" from a collection
• Questions asking for "A then B" or "first A, then B" (order matters)
• Problems involving cards from a deck, balls from an urn, or items from a bag
• Any scenario where taking one item changes the total for future draws