Problem 33
Question
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 is the probability that the Black Hawks win their next game?
Step-by-Step Solution
Verified Answer
a) The odds against are 1:3. b) The probability is \( \frac{3}{4} \).
1Step 1: Understanding Odds in Favor
The given odds in favor of the Black Hawks winning are 3 to 1. This means that for every 3 times the Black Hawks win, they are expected to lose 1 time.
2Step 2: Calculate Odds Against
To find the odds against the Black Hawks winning, you need to invert the given odds. If the odds in favor are 3:1, the odds against are the inverse, which is 1:3.
3Step 3: Calculate the Probability of Winning
Probability of an event is calculated as the ratio of favorable outcomes to the total number of outcomes. Here, the favorable outcomes are 3 (winning) and the unfavorable outcomes are 1 (losing). The total number of outcomes is 3 + 1 = 4. Therefore, the probability that the Black Hawks win their next game is given by: \[ P(\text{win}) = \frac{3}{4} \]
Key Concepts
Calculating ProbabilityUnderstanding OddsFavorable vs Unfavorable OutcomesEvents in Probability
Calculating Probability
Probability is a way to measure the chance or likelihood of an event happening. It is represented by a number between 0 and 1.
Zero means the event cannot happen, while one means the event is certain to happen.
To calculate the probability of an event, we use the formula:
\[ P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} \]
In the exercise, the Black Hawks have a 3 to 1 chance in favor of winning. This means there are 3 favorable outcomes (they win) and 1 unfavorable outcome (they lose). By adding these, we get the total number of possible outcomes, which is 4.
So, the probability of the Black Hawks winning is:
\[ P(\text{win}) = \frac{3}{4} = 0.75 \]
This means there is a 75% chance the Black Hawks will win their next game.
Zero means the event cannot happen, while one means the event is certain to happen.
To calculate the probability of an event, we use the formula:
\[ P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} \]
In the exercise, the Black Hawks have a 3 to 1 chance in favor of winning. This means there are 3 favorable outcomes (they win) and 1 unfavorable outcome (they lose). By adding these, we get the total number of possible outcomes, which is 4.
So, the probability of the Black Hawks winning is:
\[ P(\text{win}) = \frac{3}{4} = 0.75 \]
This means there is a 75% chance the Black Hawks will win their next game.
Understanding Odds
Odds compare the likelihood of one event happening to the likelihood of it not happening.
When we say the odds are 3 to 1 in favor, it means for every 3 chances of winning, there is 1 chance of losing.
Odds in favor and odds against an event are related but not the same.
If the odds in favor are 3 to 1, then the odds against are simply the inverse: 1 to 3.
This conversion gives us a clear view of both sides of the equation and makes it easier to understand the likelihood of an event happening or not happening.
When we say the odds are 3 to 1 in favor, it means for every 3 chances of winning, there is 1 chance of losing.
Odds in favor and odds against an event are related but not the same.
If the odds in favor are 3 to 1, then the odds against are simply the inverse: 1 to 3.
This conversion gives us a clear view of both sides of the equation and makes it easier to understand the likelihood of an event happening or not happening.
Favorable vs Unfavorable Outcomes
In probability, we often talk about favorable and unfavorable outcomes.
Favorable outcomes are the events we are interested in – the ones we consider a 'success'.
Unfavorable outcomes are the events we consider 'failures'.
In the given exercise, a favorable outcome is the Black Hawks winning the game. There are 3 favorable outcomes.
The unfavorable outcome is the Black Hawks losing the game. There is 1 unfavorable outcome.
Understanding the difference between these types of outcomes is crucial when calculating probability and comparing odds.
It helps to clearly define what you are measuring and what counts as a success or failure.
Favorable outcomes are the events we are interested in – the ones we consider a 'success'.
Unfavorable outcomes are the events we consider 'failures'.
In the given exercise, a favorable outcome is the Black Hawks winning the game. There are 3 favorable outcomes.
The unfavorable outcome is the Black Hawks losing the game. There is 1 unfavorable outcome.
Understanding the difference between these types of outcomes is crucial when calculating probability and comparing odds.
It helps to clearly define what you are measuring and what counts as a success or failure.
Events in Probability
An event in probability is any outcome or set of outcomes from some random process.
When calculating probability, we often look at the total number of possible outcomes and how many ways our event of interest can happen.
For the Black Hawks, the event of them winning the game happens in 3 out of 4 possible outcomes.
Events can be independent, where the outcome of one event does not affect the outcome of another, or dependent, where the outcome of one event affects the outcome of another.
In our example, each game is an independent event – the outcome of one game does not influence the outcome of the next.
When calculating probability, we often look at the total number of possible outcomes and how many ways our event of interest can happen.
For the Black Hawks, the event of them winning the game happens in 3 out of 4 possible outcomes.
Events can be independent, where the outcome of one event does not affect the outcome of another, or dependent, where the outcome of one event affects the outcome of another.
In our example, each game is an independent event – the outcome of one game does not influence the outcome of the next.
Other exercises in this chapter
Problem 32
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