Problem 20 - Algebra Form and Function

Question

There are 54 M\&Ms in a packet: 14 blue, 4 brown, 6 green, 14 orange, 7 red, and 9 yellow. (a) For each color, find the probability, as a percentage, of randomly picking that color from the packet. (b) Find the probability, as a percentage, of randomly picking a blue if someone has eaten all the reds.

Step-by-Step Solution

Answer

Answer: In general, the probabilities for each color are as follows: - Blue: 25.93% - Brown: 7.41% - Green: 11.11% - Orange: 25.93% - Red: 12.96% - Yellow: 16.67% After removing all red M&Ms, the probability of selecting a blue M&M is 29.79%.

Step 1: Calculate the probability of selecting each color in general

We can compute the probability for each color by dividing the number of M&Ms of that color by the total number of M&Ms (54). After that, we will convert the probability to percentage by multiplying it by 100: Blue: (14/54) x 100 Brown: (4/54) x 100 Green: (6/54) x 100 Orange: (14/54) x 100 Red: (7/54) x 100 Yellow: (9/54) x 100

Step 2: Calculate the probability of selecting each color as a percentage

Now we compute the percentages for each color: Blue: (14/54) x 100 = 25.93% Brown: (4/54) x 100 = 7.41% Green: (6/54) x 100 = 11.11% Orange: (14/54) x 100 = 25.93% Red: (7/54) x 100 = 12.96% Yellow: (9/54) x 100 = 16.67% For part (a), the probabilities for each color as percentages are as follows: Blue: 25.93% Brown: 7.41% Green: 11.11% Orange: 25.93% Red: 12.96% Yellow: 16.67%

Step 3: Find the new total of M&Ms after removing reds

If all the red M&Ms are removed, we need to find the new total number of M&Ms left: New total = Initial total - Number of reds = 54 - 7 = 47 M&Ms

Step 4: Calculate the probability of selecting blue after reds are removed

Now that we have the new total of M&Ms after removing the red ones, we can calculate the probability of selecting a blue M&M as a percentage: Blue (after removing reds): (14/47) x 100 = 29.79% For part (b), the probability of randomly picking a blue M&M after all red M&Ms are removed is 29.79%.

Key Concepts

Understanding Percentage in Probability

Percentages are a way of expressing a number as a part of a whole. In probability, we often convert fractions into percentages to easily understand the likelihood of an event occurring. To convert a probability into a percentage:

• Divide the number of favorable outcomes by the total number of possible outcomes to obtain a fraction.
• Multiply the fraction by 100 to convert it into a percentage.

For instance, the probability of choosing a blue M&M from a packet is initially a fraction: \( \frac{14}{54} \). When converted into a percentage, it gives \( 25.93\% \).
Percentages provide a clear and intuitive way to gauge how likely an outcome is, which is why they're useful in probability discussions.

Random Selection explained

Random selection refers to choosing an item from a set without any preference or pattern. This process ensures that each item has an equal chance of being selected.
In the context of the M&M example provided, if you were to randomly select an M&M from the packet, each color would have a specific probability based on its relative quantity in the packet. Calculating this involves:

• Counting the individual items.
• Dividing by the total number of items in the set.

This establishes a fair chance for each M&M based on its color, which is vital in many real-world applications where unbiased selection is crucial.

Diving into Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It's a core concept in probability that helps understand how probabilities change when conditions are modified.
In the M&M problem, after all the red M&Ms are eaten, the probability of picking a blue M&M changes. Originally calculated with the full packet, removing all the reds changes the total number of M&Ms, which alters the probability.
The new probability for blue M&Ms is calculated as:

• Blue M&Ms remain the same in number, 14.
• The total number of M&Ms now is 47 (after reds are removed).
• This gives a new probability fraction: \( \frac{14}{47} \)
• Convert this fraction to a percentage: \( 29.79\% \).

Thus, conditional probability reveals how outcomes are dependent on pre-existing conditions under which they occur.