Problem 14 - Algebra and Trigonometry

Question

\(13-14\) . A ball is drawn randomly from a jar that contains five red balls, two white balls, and one yellow ball. Find the probability of the given event. (a) Neither a white nor yellow ball is drawn. (b) A red, white, or yellow ball is drawn. (c) The ball that is drawn is not white.

Step-by-Step Solution

Answer

(a) \(\frac{5}{8}\), (b) 1, (c) \(\frac{3}{4}\)

Step 1: Determine Total Number of Balls

First, determine the total number of balls in the jar. There are 5 red balls, 2 white balls, and 1 yellow ball. Therefore, the total number of balls is \(5 + 2 + 1 = 8\).

Step 2: Calculate Probability for Neither a White nor Yellow Ball (Part a)

For neither a white nor yellow ball to be drawn, only a red ball can be drawn. There are 5 red balls. Therefore, the probability is \(\frac{5}{8}\) since there are 5 successful outcomes and 8 possible outcomes.

Step 3: Calculate Probability for Drawing a Red, White, or Yellow Ball (Part b)

The event of drawing a red, white, or yellow ball includes all possibilities because all balls are either red, white, or yellow. Thus, the probability is \(\frac{8}{8} = 1\).

Step 4: Calculate Probability the Drawn Ball is Not White (Part c)

For a ball not to be white, it must be either red or yellow. There are 5 red balls and 1 yellow ball, totaling 6 balls that are not white. The probability is \(\frac{6}{8} = \frac{3}{4}\).

Key Concepts

probability theory

Probability theory is the branch of mathematics that deals with calculating the likelihood of different outcomes. It's all about understanding and working with the uncertainties in events.
In this context, we have various outcomes for the event of drawing a ball from the jar, and we're trying to establish how likely each outcome is.

• Clarity is key in probability theory. We need precise definitions of each possible event.
• The sum of the probabilities of all possible outcomes of an event equals 1.
• The probability of an impossible event is 0.

In this exercise, the entire jar and its contents represent our 'sample space,' the set of all possible outcomes (8 balls in total). Each ball's chance of being drawn relates directly to the principles of probability theory.

probability of events

The probability of events refers to the likelihood of specific outcomes occurring within a sample space. For basic events, it measures how many times certain outcomes can happen compared to all possible outcomes.
Events can be more complex, such as those consisting of several combined outcomes, like drawing either a red, white, or yellow ball.
In this exercise, let's examine:

• "Neither a white nor yellow ball is drawn" focuses on red balls only since red is not white or yellow. The event of drawing red is accountable by the probability \(\frac{5}{8}\).
• "A red, white, or yellow ball is drawn" encompasses all possible outcomes in the jar. Here, the probability becomes \(\frac{8}{8} = 1\) because any drawn ball meets the event's criteria.
• "The ball that is drawn is not white," includes both red and yellow, giving a probability of \(\frac{6}{8} = \frac{3}{4}\).

Understanding these events sharpens our grasp on distinguishing among varied outcomes based on set conditions.

calculation of probability

Calculation of probability involves using the ratio of favorable outcomes to the total number of possible outcomes. This method gives a clear mathematical expression of how likely an event is to happen.
For any event, the probability \(P\) is determined by
\[P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]Let's apply this calculation:

• For the event "neither a white nor yellow ball is drawn": There are 5 red balls, so \(\frac{5}{8}\) represents the probability.
• For "a red, white, or yellow ball is drawn": Every ball would suffice, hence \(\frac{8}{8} = 1\).
• For "the ball drawn is not white": 5 red and 1 yellow make 6 non-white balls, with the probability \(\frac{6}{8} = \frac{3}{4}\).

This straightforward mathematical process allows us to quantify how likely each event is relative to others within the same sample space.