10 Drill Questions: Bayesian reasoning
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NoteB. Odds and probabilities
Drill 10. 1 There are two outcomes possible: Win or Lose. Your personal opinion is that the relative probabilities of Win and Lose are 15 and 25, respectively.
- What’s the odds of Win?
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- What’s the odds of Lose?
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Drill 10. 2 There are only two horses in the race: Valentine and Epitaph. Valentine’s odds are 9 to 6.
- What are Epitaph’s odds?
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- Which of these is a correct statement of the relative probabilities for Valentine and Epitaph respectively?
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Drill 10. 3 It may seem strange that mathematics has two different but equivalent ways of quantifying how likely a given outcome is from a random event: absolute probability and odds.
To understand the equivalence, consider the possible outcome A from a random event. Suppose this has absolute probability of \(p\) and odds of \(d\). These two formulas show how to calculate one from the other.
\[\text{absolute probability}\ \ p = \frac{d}{1+d}\ \ \ \ \ \text{and}\ \ \ \ \ \text{odds}\ \ d = \frac{p}{1-p}\]
- If the odds of A is 2, what is the absolute probability of A?
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- If the absolute probability of A is 50%, what is the odds of A?
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- Which of these ranges expresses the highest and lowest possible values for an absolute probability?
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- Which of these ranges expresses the highest and lowest possible values for an odds? (Hint: plug in the ends of the correct range from (3) into the formula for odds.)
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