Problem 54 - Algebra 2

Question

Use this information for Exercises \(53-58\) . Bag 1 contains 5 red marbles, 1 blue marble, 3 yellow marbles, and 2 green marbles. Bag 2 contains 1 red pencil, 3 red pens, 2 blue pencils, and 5 blue pens. One marble is drawn from bag \(1 .\) What is the probability that the marble is blue or not green?

Step-by-Step Solution

Answer

The probability that the marble is blue or not green is \(10/11\).

Step 1: Compute Total Number of Outcomes

First calculate the total number of outcomes. This is simply the total number of marbles in the bag, which is \(5 + 1 + 3 + 2 = 11\).

Step 2: Compute Number of Favorable Outcomes for Blue Marble

Count the total number of blue marbles in the bag, which is 1. This is the number of favorable outcomes for the event 'the draw is blue'.

Step 3: Compute Number of Favorable Outcomes for Not Green

The number of 'not green' outcomes is the total number of marbles minus the number of green marbles. There are 2 green marbles, so the number of non-green outcomes is \(11 - 2 = 9\).

Step 4: Compute Probability of Each Event

Compute the probability of each event. The probability that the drawn marble is blue is \(1/11\), and the probability that it is not green is \(9/11\).

Step 5: Use the Rule of Addition for Mutually Exclusive Events

Since these two events are mutually exclusive (a marble cannot be both blue and not green at the same time), the probability that the drawn marble is blue or not green can be found by adding the probabilities computed in Step 4: \(1/11 + 9/11 = 10/11\).

Key Concepts

Understanding Mutually Exclusive Events

In probability, the term "mutually exclusive events" refers to events that cannot both occur at the same time. Here, if you have two events and one event happens, the other event cannot happen simultaneously.
For example, in the context of drawing marbles, if an event is drawing a blue marble from the bag, it cannot also be a green marble. These two outcomes are mutually exclusive because you cannot draw one marble that is both blue and green.
Mutually exclusive events help simplify probability calculations, as they allow us to directly combine probabilities of different events without concern for overlap.

Applying the Addition Rule

The addition rule is vital in calculating the probability of two or more mutually exclusive events. It states that if two events, A and B, are mutually exclusive, the probability of A or B occurring is simply the sum of their individual probabilities.
This means that for events that have no overlap, you can add the probabilities directly to get the combined outcome probability.
In our example, the marble being blue (Event A) and the marble being not green (Event B) are mutually exclusive. Thus, the probability of drawing a marble that is either blue or not green is the sum of the probabilities of these individual events: \(\frac{1}{11} + \frac{9}{11} = \frac{10}{11}\).
This technique simplifies calculations and is extremely helpful in situations involving clearly separate outcomes.

Outcome Count in Probability

Outcome count refers to the total number of possible results when an action is performed. It is a fundamental part of calculating probabilities, as it helps in determining both the total possible outcomes and the favorable outcomes for an event.
In the marble example, understanding the outcome count starts with acknowledging there are 11 marbles in total. This number represents all the potential outcomes when a marble is drawn from the bag.
Accurately counting outcomes is crucial as it forms the base denominator in probability calculations, allowing for precise determination of an event's probability.

Identifying Favorable Outcomes

Favorable outcomes are the results that fulfill the condition you are calculating the probability for. Knowing how to count these correctly is as essential as counting total outcomes.
For an event like drawing a blue marble, the only favorable outcome is picking the one blue marble available, which gives us one favorable outcome.
On the other hand, determining non-green outcomes means excluding the unfavorable results (the two green marbles in this case), leading to a total of nine non-green outcomes from the eleven marbles.

• Favorable outcomes for blue: 1
• Favorable outcomes for not green: 9

Understanding favorable outcomes helps you identify specific results that satisfy the condition of your probability question, providing clarity and focus during calculations.