Understanding Marble Probability

This problem asks us to find the probability of drawing a black or a blue marble from a glass jar containing marbles of different colors. Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

Calculating Total Possible Outcomes

First, let's find the total number of marbles in the jar. The jar contains:

• 6 white marbles
• 8 black marbles
• 4 red marbles
• 3 blue marbles

The total number of marbles is the sum of the marbles of all colors:

Total marbles = Number of white + Number of black + Number of red + Number of blue

Total marbles = \(6 + 8 + 4 + 3 = 21\)

So, there are 21 possible outcomes when selecting a single marble from the jar.

Identifying Favorable Outcomes (Black or Blue)

We are interested in the probability of drawing a marble that is either black or blue. These are our favorable outcomes. The number of favorable outcomes is the total number of black marbles plus the total number of blue marbles, because a marble cannot be both black and blue at the same time (these events are mutually exclusive).

Number of black marbles = 8

Number of blue marbles = 3

Number of favorable outcomes (black or blue) = Number of black marbles + Number of blue marbles

Number of favorable outcomes = \(8 + 3 = 11\)

Calculating the Probability

The probability of an event is given by the formula:

P(Event) = \(\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)

In this case, the event is drawing a black or blue marble.

P(Black or Blue) = \(\frac{\text{Number of black or blue marbles}}{\text{Total number of marbles}}\)

P(Black or Blue) = \(\frac{11}{21}\)

Therefore, the probability of choosing a black or blue marble is \(\frac{11}{21}\).

Summary of Probability Calculation

• Total marbles in the jar = 21
• Number of black marbles = 8
• Number of blue marbles = 3
• Number of black or blue marbles = \(8 + 3 = 11\)
• Probability (Black or Blue) = \(\frac{11}{21}\)

Revision Table: Key Probability Concepts

Additional Information: Probability Rules

When dealing with probabilities, especially involving multiple events, it's important to understand key rules:

• Addition Rule for Mutually Exclusive Events: As used in this problem, if two events A and B are mutually exclusive (cannot occur together), the probability that either A or B occurs is the sum of their individual probabilities: \(P(A \text{ or } B) = P(A) + P(B)\).
  • In our case, drawing a black marble (Event A) and drawing a blue marble (Event B) are mutually exclusive.
  • \(P(\text{Black}) = \frac{8}{21}\)
  • \(P(\text{Blue}) = \frac{3}{21}\)
  • \(P(\text{Black or Blue}) = P(\text{Black}) + P(\text{Blue}) = \frac{8}{21} + \frac{3}{21} = \frac{8+3}{21} = \frac{11}{21}\). This confirms our calculation.
• Addition Rule for Non-Mutually Exclusive Events: If two events A and B are not mutually exclusive (can occur together), the probability that either A or B occurs is the sum of their individual probabilities minus the probability of both occurring: \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\). This rule was not needed here because drawing a black and a blue marble simultaneously in a single draw is impossible.
• Multiplication Rule for Independent Events: If two events A and B are independent (the outcome of one doesn't affect the other), the probability that both A and B occur is the product of their individual probabilities: \(P(A \text{ and } B) = P(A) \times P(B)\). This is typically used for sequential events with replacement or events from different sources.

Understanding these rules helps in solving various probability problems, whether dealing with marbles, cards, dice, or other random events.