Problem 21 - A Graphical Approach to College Algebra

Question

Concept Check \(\quad\) Match each probability in parts (a)-(g) with one of the statements in A-F. (a) \(P(E)=-0.1\) (b) \(P(E)=0.01\) (c) \(P(E)=1\) (d) \(P(E)=2\) (e) \(P(E)=0.99\) (f) \(P(E)=0\) (g) \(P(E)=0.5\) A. The event is certain to occur. B. The event is impossible. C. The event is very likely to occur. D. The event is very unlikely to occur. E. The event is just as likely to occur as not occur. F. This probability cannot occur.

Step-by-Step Solution

Answer

(a) F, (b) D, (c) A, (d) F, (e) C, (f) B, (g) E.

Step 1: Understanding Probability Concepts

Probability values must be between 0 and 1, inclusive. Values outside this range, like negative probabilities or those greater than 1, are invalid.

Step 2: Probability: Negative or Greater than 1

Probabilities (a) \( P(E) = -0.1 \) and (d) \( P(E) = 2 \) fall outside the valid range of 0 to 1. Therefore, these probabilities cannot occur, matching Statement F.

Step 3: Probability: Certainty

For (c) \( P(E) = 1 \), the probability of the event is exactly 1, meaning the event is certain to occur, which matches Statement A.

Step 4: Probability: Impossibility

For (f) \( P(E) = 0 \), the probability is 0, meaning the event is impossible, corresponding to Statement B.

Step 5: Probability: Very Likely

The probability in (e) \( P(E) = 0.99 \) is close to 1, indicating the event is very likely to occur, matching Statement C.

Step 6: Probability: Very Unlikely

In (b) \( P(E) = 0.01 \), the probability is very small, suggesting the event is very unlikely to occur, which is Statement D.

Step 7: Probability: Equally Likely

For (g) \( P(E) = 0.5 \), the event is just as likely to occur as not, matching Statement E.

Key Concepts

Probability Range

Probability measures the chance of an event occurring, and its value must always lie within a specific range. This range is

• From 0, representing an impossible event
• To 1, representing a certain event

Probabilities such as negative numbers or numbers greater than 1 are not valid. They do not make sense because they do not fit within the defined range.
Instead, we use a range from 0 to 1 to ensure that our understanding of the likelihood of events is consistent and logical.
It's important to remember that any probability value below 0 or above 1 cannot occur. By maintaining this range, we are guaranteed to have meaningful interpretations of probabilities, providing clarity about the likelihood of events.

Certain and Impossible Events

Understanding certain and impossible events is key to grasping the limits of probability. When an event has a probability of 0, it is categorized as impossible.
This means there's absolutely no chance of the event happening. Conversely, when the probability is 1, the event is certain, meaning it will definitely occur.
These extreme ends of the probability range help us make clear judgments about events.
For instance:

• If you flip a coin expecting it to land on its edge, the probability is 0 because it's practically impossible.
• If you roll a fair die, the probability that the outcome will be a number between 1 and 6 is 1, as it is a certainty.

By clearly defining these boundaries, we can distinguish between events that have no chance and those that are guaranteed.

Likelihood Interpretation

The interpretation of likelihood plays a vital role in understanding probabilities between the extremes of 0 and 1. Probabilities help us predict how likely it is for an event to happen.
Here's how to interpret different levels:

• A probability close to 0 (but not 0) indicates an event is very unlikely to occur. For example, a probability of 0.01 suggests that an event, while possible, is highly unlikely.
• A probability near 1 (but not 1) indicates an event is very likely to occur. A value of 0.99 would imply the event is almost certain to happen.
• When the probability is exactly 0.5, it means the event is equally likely or unlikely to occur. This is like a 50-50 chance, similar to flipping a fair coin.

Recognizing these gradations helps us quantify the certainty or uncertainty associated with real-life occurrences. By understanding these nuances, we can make informed predictions about the likelihood of various events.