Problem 32 - Algebra for College Students

Question

If the probability of a coal miners' strike this year is \(\frac{9}{10}\) then what are the odds a) in favor of a strike? b) against a strike?

Step-by-Step Solution

Answer

a) 9 to 1 b) 1 to 9

Step 1 - Understanding Probability

The probability of a coal miners' strike is given as \(\frac{9}{10}\). This means there is a \(\frac{9}{10}\) chance a strike will happen.

Step 2 - Converting Probability to Odds in Favor

To find the odds in favor of an event happening, use the formula: \(\text{Odds in favor} = \frac{P(\text{event})}{1 - P(\text{event})}\). Here, \(\frac{9}{10}\) is the probability of the strike happening, so the odds in favor are \(\frac{9/10}{1 - 9/10} = \frac{9/10}{1/10} = 9\). This means the odds in favor of the strike are 9 to 1.

Step 3 - Converting Probability to Odds Against

To find the odds against an event happening, use the formula: \(\text{Odds against} = \frac{1 - P(\text{event})}{P(\text{event})}\). Here, \(\frac{9}{10}\) is the probability of the strike happening, so the odds against are \(\frac{1 - 9/10}{9/10} = \frac{1/10}{9/10} = \frac{1}{9}\). This means the odds against the strike are 1 to 9.

Key Concepts

probability

Probability is a measure of the likelihood that a certain event will occur. It ranges from 0 to 1, with 0 meaning an event will not happen and 1 meaning it will certainly happen. For example, a probability of \(\frac{9}{10}\) means there is a very high chance, specifically 90%, that the event will occur.

The formula for probability is given by:
\ P(event) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \

Let's break it down more:

• If an event is certain, like the sun rising tomorrow, the probability is 1.
• If an event is impossible, like pigs flying without aid, the probability is 0.
• A probability of 0.5 means there's a 50-50 chance of the event occurring, like flipping a fair coin and getting heads.

The probability helps us quantify uncertain situations and make more informed decisions.

odds in favor

When we talk about the 'odds in favor' of an event, we are comparing the probability that the event will happen to the probability that it will not happen. If the probability of an event is \(\frac{9}{10}\), the formula to calculate the odds in favor is: \ \text{Odds in favor} = \frac{P(\text{event})}{1 - P(\text{event})} \

Plugging in our values: \ \text{Odds in favor} = \frac{\frac{9}{10}}{1 - \frac{9}{10}} = \frac{\frac{9}{10}}{\frac{1}{10}} = 9 \
This means the odds in favor of the coal miners' strike are 9 to 1. In simpler terms, for every 9 chances that the strike will happen, there is 1 chance it won't. Odds in favor are useful for understanding and communicating the likelihood of an event happening in a different way than probability.

odds against

The 'odds against' an event are the opposite of the 'odds in favor'. They compare the probability that the event will not happen to the probability that it will happen. Using the same probability of \(\frac{9}{10}\) for the event, the formula to calculate the odds against is: \ \text{Odds against} = \frac{1 - P(\text{event})}{P(\text{event})} \

Inserting the values, we get:
\ \text{Odds against} = \frac{1 - \frac{9}{10}}{\frac{9}{10}} = \frac{\frac{1}{10}}{\frac{9}{10}} = \frac{1}{9} \
This means the odds against the coal miners' strike are 1 to 9. So for every 1 chance that the strike won't happen, there are 9 chances that it will.

Understanding odds against helps in risk assessment and in making more calculated decisions, especially when the probability of an event is low.
Both 'odds in favor' and 'odds against' are handy for representing risks and chances in real-world scenarios.