Questions: Suppose the probability of an event is 20/27. What are the odds for the event happening? to What are the odds against the event happening? to

Suppose the probability of an event is 20/27. What are the odds for the event happening? to What are the odds against the event happening? to
Transcript text: prod.ccsnh.edu/courses/86834/assignments/1653087 Due Sunday by 11:59pm Points 100 Submitting an external too Chapter 8: Introduction to Probability - Hom Score: 61/100 21/31 answered Question 22 Suppose the probability of an event is $\frac{20}{27}$. What are the odds for the event happening? $\square$ to $\square$ What are the odds against the event happening? $\square$ to $\square$ Question Help: Video Submit Question
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Solution

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Solution Steps

To solve the problem, we need to understand the relationship between probability and odds. The probability of an event happening is given as \(\frac{20}{27}\). The odds for the event happening are calculated as the ratio of the probability of the event happening to the probability of the event not happening. Similarly, the odds against the event happening are the inverse of the odds for the event happening.

Solution Approach
  1. Calculate the probability of the event not happening.
  2. Calculate the odds for the event happening.
  3. Calculate the odds against the event happening.
Step 1: Calculate the Probability of the Event Not Happening

Given the probability of the event happening is

\[ P(A) = \frac{20}{27} \]

the probability of the event not happening is

\[ P(A') = 1 - P(A) = 1 - \frac{20}{27} = \frac{7}{27}. \]

Step 2: Calculate the Odds for the Event Happening

The odds for the event happening can be calculated using the formula:

\[ \text{Odds for} = \frac{P(A)}{P(A')} = \frac{\frac{20}{27}}{\frac{7}{27}} = \frac{20}{7} \approx 2.8571. \]

This can be expressed as

\[ \text{Odds for} \approx 2.86 \text{ to } 1. \]

Step 3: Calculate the Odds Against the Event Happening

The odds against the event happening are the inverse of the odds for the event happening:

\[ \text{Odds against} = \frac{P(A')}{P(A)} = \frac{\frac{7}{27}}{\frac{20}{27}} = \frac{7}{20} \approx 0.35. \]

This can be expressed as

\[ \text{Odds against} \approx 0.35 \text{ to } 1. \]

Final Answer

The odds for the event happening are approximately \( \boxed{2.86 \text{ to } 1} \) and the odds against the event happening are approximately \( \boxed{0.35 \text{ to } 1} \).

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