Problem 4 - Algebra 2

Question

There are 8 action, 3 comedy, and 5 children's DVDs on a shelf. Suppose two DVDs are selected at random from the shelf. Find each probability. \(P(2 \text { action } \mathrm{DVDs}),\) if no replacement occurs

Step-by-Step Solution

Answer

The probability is \(\frac{7}{30}\).

Step 1: Determine Total Number of DVDs

First, count the total number of DVDs on the shelf. There are 8 action, 3 comedy, and 5 children's DVDs. Thus, \(8 + 3 + 5 = 16\) total DVDs.

Step 2: Calculate Ways to Select 2 Action DVDs

We need the number of ways to choose 2 action DVDs from the 8 available. This is given by the combination formula \(\binom{n}{r}\), which in this case is \(\binom{8}{2}\). Calculate this: \(\binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28\).

Step 3: Calculate Ways to Select Any 2 DVDs

Now find the number of ways to select any 2 DVDs from the total of 16 DVDs. Use the same combination formula: \(\binom{16}{2}\). Calculate this: \(\binom{16}{2} = \frac{16 \times 15}{2 \times 1} = 120\).

Step 4: Compute Probability of Selecting 2 Action DVDs

The probability of selecting 2 action DVDs is the number of ways to select 2 action DVDs divided by the number of ways to select any 2 DVDs. Thus, the probability is: \(P(2 \text{ action DVDs}) = \frac{\binom{8}{2}}{\binom{16}{2}} = \frac{28}{120}\).

Step 5: Simplify the Probability

Simplify the fraction \(\frac{28}{120}\) by dividing the numerator and the denominator by their greatest common divisor, which is 4. This gives \(\frac{28}{120} = \frac{7}{30}\).

Key Concepts

Combination

Combinations are a fundamental part of probability and statistics. They allow us to determine the number of ways to select items from a larger pool without regard to the order in which the items are selected. This is particularly useful when we are calculating probabilities.

• The formula for a combination is given by: \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where:
  • \(n\) is the total number of items in the pool,
  • \(r\) is the number of items to choose,
  • \(!\) denotes factorial, which is the product of all positive integers up to that number.
• In our DVD example, to choose 2 action DVDs from 8, we set \(n = 8\) and \(r = 2\) which simplifies the problem to: \[\binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28\]

Each step helps in solving problems related to combinations, ensuring precision and reducing possible errors.

Probability without Replacement

Probability without replacement refers to situations where items are not put back into the pool after being selected, changing the subsequent probability calculations. This impacts the total number of possible outcomes and needs a thoughtful approach.

• In our DVD example, once we select an action DVD, it isn't replaced, so the total number of DVDs becomes 15 (one less than before)
• The probability of specific sequences of events is calculated by multiplying probabilities from each step, adjusting for the changed number of total items.
• The probability of selecting two action DVDs without replacement is calculated as:
  • First action DVD: \(\frac{8}{16}\)
  • Second action DVD: \(\frac{7}{15}\) (since one action DVD has already been taken out)
  • This gives a combined probability of: \(\frac{8}{16} \times \frac{7}{15} = \frac{56}{240} = \frac{7}{30}\) after simplification.

Understanding "without replacement" is crucial. It changes the dynamic and specific approaches are needed to effectively handle these calculations in probability.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics, as it presents the result in a more understandable form. This involves reducing the fraction to its simplest form by dividing the numerator and the denominator by their greatest common divisor (GCD).

• In our example with the DVD probabilities, the step involved simplifying \(\frac{28}{120}\).
• First, we need to find the GCD of 28 and 120. Here, this is 4.
• Divide both the numerator and the denominator by their GCD:\[\frac{28}{120} = \frac{28 \div 4}{120 \div 4} = \frac{7}{30}\]

This conversion to simpler terms makes the fraction more intuitive to interpret and useful in subsequent calculations. Practicing this skill ensures accuracy in presenting mathematical results.