Problem 34
Question
If the odds are 5 to 1 against the Democratic presidential nominee winning the election, then a) what are the odds in favor of the Democrat winning the election? b) what is the probability that the Democrat wins the election?
Step-by-Step Solution
Verified Answer
a) 1 to 5, b) \( \frac{1}{6} \)
1Step 1: Understand the given odds
The problem states that the odds are 5 to 1 against the Democratic nominee. This means that for every 6 outcomes (5 against, 1 in favor), 5 are against and 1 is in favor.
2Step 2: Convert odds against to odds in favor
To find the odds in favor of the Democrat winning, we simply reverse the ratio. If the odds are 5 to 1 against, then the odds in favor are 1 to 5.
3Step 3: Calculate the probability
The probability is calculated by taking the number of favorable outcomes and dividing it by the total number of possible outcomes. Given odds of 1 to 5 in favor, the total number of possible outcomes is 1 (favorable) + 5 (against) = 6. Therefore, the probability (P) is calculated as: \[ P = \frac{1}{6} \]
Key Concepts
Odds AgainstProbability CalculationOdds in Favor
Odds Against
Understanding 'odds against' is crucial in probability and statistics. When we say the odds are against something, we mean that there are more outcomes where the event does not happen compared to when it does happen.
For example, if the odds are 5 to 1 against the Democratic presidential nominee winning, it means there are 5 outcomes where they lose for every 1 outcome where they win.
In this scenario:
For example, if the odds are 5 to 1 against the Democratic presidential nominee winning, it means there are 5 outcomes where they lose for every 1 outcome where they win.
In this scenario:
- There are 5 outcomes against (losing).
- There is 1 outcome in favor (winning).
Probability Calculation
Calculating probability is straightforward once you comprehend the basics of odds.
Probability is the measure of how likely an event is to occur compared to the total number of possible outcomes.
To find the probability, we use the formula:
\[ P = \frac{Number\; of\; Favorable\; Outcomes}{Total\; Number\; of\; Possible\; Outcomes} \]
For our exercise:
We first need to find the total number of possible outcomes, which is 6 (since 5 + 1 = 6).
The number of favorable outcomes is 1 (only one win scenario).
So, the probability (P) of the Democrat winning is:
\[ P = \frac{1}{6} \]
This fraction tells us that there's a one in six chance of winning.
Probability is the measure of how likely an event is to occur compared to the total number of possible outcomes.
To find the probability, we use the formula:
\[ P = \frac{Number\; of\; Favorable\; Outcomes}{Total\; Number\; of\; Possible\; Outcomes} \]
For our exercise:
We first need to find the total number of possible outcomes, which is 6 (since 5 + 1 = 6).
The number of favorable outcomes is 1 (only one win scenario).
So, the probability (P) of the Democrat winning is:
\[ P = \frac{1}{6} \]
This fraction tells us that there's a one in six chance of winning.
Odds in Favor
Understanding the concept of 'odds in favor' helps us interpret the likelihood of a particular event occurring.
If initially the odds are given as 'against' an event, you can easily convert it to 'in favor' by reversing the ratio.
In our example, the odds were given as 5 to 1 against the Democrat. To find the odds in favor, we swap the numbers around, making it 1 to 5.
This tells us for every instance the Democrat wins, there are 5 instances where they lose.
Remember, the 'odds in favor' ratio allows us to see how many successful outcomes there are relative to the unsuccessful ones, flipped from the 'odds against' perspective.
If initially the odds are given as 'against' an event, you can easily convert it to 'in favor' by reversing the ratio.
In our example, the odds were given as 5 to 1 against the Democrat. To find the odds in favor, we swap the numbers around, making it 1 to 5.
This tells us for every instance the Democrat wins, there are 5 instances where they lose.
Remember, the 'odds in favor' ratio allows us to see how many successful outcomes there are relative to the unsuccessful ones, flipped from the 'odds against' perspective.
Other exercises in this chapter
Problem 33
If the odds are 3 to 1 in favor of the Black Hawks winning their next game, then a) what are the odds against the Black Hawks winning their next game? b) what i
View solution Problem 34
Make a list of all of the permutations of the letters \(A, B, C\) \(\mathrm{D},\) and \(\mathrm{E}\) taken 3 at a time. How many permutations should be in your
View solution Problem 35
Evaluate expression. \(4 !\)
View solution Problem 35
What are the odds in favor of getting exactly 2 heads in 3 tosses of a coin?
View solution