Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: a bag of M\&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M\&Ms. Reaching into the bag, a person grabs 5 M\&Ms. What is the probability of getting all blue M\&Ms?

Step-by-Step Solution

Answer

The probability is approximately 0.000462.

Step 1: Understand the total number of M&Ms

First, we need to determine the total number of M&Ms in the bag by adding up all the different colored M&Ms: 12 (blue) + 6 (brown) + 10 (orange) + 8 (yellow) + 8 (red) + 4 (green). The total is 48 M&Ms.

Step 2: Determine the number of favorable outcomes

We want to find the number of ways to select 5 blue M&Ms from the 12 blue M&Ms available. This is a combination problem, calculated as \( \binom{12}{5} \).

Step 3: Calculate the total number of ways to pick 5 M&Ms

Now calculate the total number of ways to pick any 5 M&Ms from the 48 available. This is also a combination problem, calculated as \( \binom{48}{5} \).

Step 4: Apply the formula for probability

The probability of selecting all blue M&Ms is the ratio of the number of favorable outcomes to the total number of outcomes: \( P(\text{all blue}) = \frac{\binom{12}{5}}{\binom{48}{5}} \).

Step 5: Compute the probabilities

Calculate \( \binom{12}{5} \) which is 792, and \( \binom{48}{5} \) which is 1,712,304. Plug these values into the probability formula: \( P(\text{all blue}) = \frac{792}{1,712,304} \approx 0.00046245 \).

Key Concepts

Combination

In probability, a *combination* is a way of selecting items from a larger pool where the order of selection does not matter. This concept is crucial when dealing with problems like selecting M&Ms from a bag because we are interested in the different groups we can form.
To calculate combinations, we use the formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from and \( r \) is the number of items we are selecting. For example, when selecting 5 blue M&Ms from 12, we use \( \binom{12}{5} \).
This formula is expressed as:
\[ \binom{n}{r} = \frac{n!}{r! \, (n-r)!} \]
Here, '!' denotes factorial, which is the product of an integer and all the integers below it. So \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Thus, combinations provide a way to mathematically determine how many different groups of items can be selected from a larger set.

Favorable outcomes

*Favorable outcomes* refer to the specific outcomes that satisfy the condition of a probability problem. These are the outcomes we are interested in when calculating a probability. In the M&Ms problem, a favorable outcome is one where all 5 M&Ms selected are blue.
To find the number of favorable outcomes, you calculate the number of ways to select 5 blue M&Ms from the 12 available. This is done using the combination formula \( \binom{12}{5} \). When calculated, it gives 792 favorable outcomes.
avorablyleveraging these outcomes is essential to finding probability, as they are used as the numerator in probability calculations. This is because probability is typically the ratio of favorable outcomes to total possible outcomes.

Total outcomes

The term *total outcomes* refers to all the possible outcomes of an event without considering any specific conditions. When you calculate probabilities, knowing the total outcomes provides the denominator in the probability fraction.
In our M&Ms example, the total outcomes refer to the number of ways to pick any 5 M&Ms from the bag of 48. This utilizes the combination formula again, \( \binom{48}{5} \), which equals 1,712,304.
Understanding total outcomes is vital because it provides a complete view of the scenario, while the comparison to favorable outcomes offers insight into the likelihood of a specific event occurring.
By comparing favorable to total outcomes, you form a probability ratio, which helps quantify the chances of any particular event, like grabbing all blue M&Ms.