91Ó°ÊÓ

Sometimes probability statements are expressed in terms of odds. The odds in favor of an event \(A\) are the ratio \(\frac{P(A)}{P(n o t A)}=\frac{P(A)}{P\left(A^{c}\right)}\) For instance, if \(P(A)=0.60,\) then \(P\left(A^{C}\right)=0.40\) and the odds in favor of \(A\) are $$ \frac{0.60}{0.40}=\frac{6}{4}=\frac{3}{2}, \text { written as } 3 \text { to } 2 \text { or } 3: 2 $$ (a) Show that if we are given the odds in favor of event \(A\) as \(n: m,\) the probability of event \(A\) is given by \(P(A)=\frac{n}{n+m} .\) Hint: Solve the equation $$ \frac{n}{m}=\frac{P(A)}{1-P(A)} \text { for } P(A) $$ (b) A telemarketing supervisor tells a new worker that the odds of making a sale on a single call are 2 to \(15 .\) What is the probability of a successful call? (c) A sports announcer says that the odds a basketball player will make a free throw shot are 3 to \(5 .\) What is the probability the player will make the shot?

Step by step solution

01

Equation Setup from Odds to Probability

Given the odds in favor of an event \(A\) as \(n:m\), the equation is \(\frac{n}{m} = \frac{P(A)}{1-P(A)}\). Our goal is to solve this equation to express \(P(A)\) in terms of \(n\) and \(m\).

02

Cross-Multiplication

To eliminate the fractions, perform cross-multiplication on the equation \(\frac{n}{m} = \frac{P(A)}{1-P(A)}\). This results in \(n(1-P(A)) = mP(A)\).

03

Distribute and Rearrange Terms

Expand the equation: \(n - nP(A) = mP(A)\). Rearrange the terms to isolate the terms involving \(P(A)\): \(n = nP(A) + mP(A)\).

04

Factor and Solve for P(A)

Factor \(P(A)\) out from the right side: \(n = P(A)(n + m)\). Solving for \(P(A)\) gives \(P(A) = \frac{n}{n + m}\). This proves part (a) of the exercise.

05

Calculate Probability for Sale - Part b

The odds of making a sale on a single call are given as 2 to 15. Using the formula \(P(A) = \frac{n}{n+m}\), substitute \(n = 2\) and \(m = 15\). Thus, \(P(A) = \frac{2}{2+15} = \frac{2}{17}\).

06

Calculate Probability for Free Throw - Part c

The odds that a player will make a free throw are given as 3 to 5. Substitute \(n = 3\) and \(m = 5\) into the formula \(P(A) = \frac{n}{n+m}\). Thus, \(P(A) = \frac{3}{3+5} = \frac{3}{8}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Odds Ratio

Odds ratio is a way of describing the likelihood of an event happening compared to it not happening. Imagine you have a bag with marbles. If the odds are 3:2, it means for every 3 marbles of one kind, you have 2 marbles of another. This ratio tells you how many times you expect the first event compared to the second.

It's often expressed as "3 to 2" or simply "3:2." This means that for each occurrence of the event, you expect it not to occur twice. This ratio can easily convert into a probability, which helps understand how likely an event is to occur.

• To convert odds to probability, use this method: if the odds in favor of an event is given as "n to m," then the probability of the event is given by the formula: \(P(A) = \frac{n}{n+m}\).

Using odds helps in fields like statistics, sports, and even health sciences to express and understand likelihoods in a more intuitive manner.

Probability Calculation

Calculating probability is a fundamental aspect of understanding how likely an event is to happen. In our example with odds, once we have the odds ratio given as "n:m," we can find the probability with a simple formula.

Here's how you navigate through this:

• Identify your odds: For example, if the odds are "2:15," this means for every successful event, there are 15 failures.
• Calculate the probability: Use \(P(A) = \frac{n}{n+m}\). Plugging the numbers in gives \(P(A) = \frac{2}{2+15} = \frac{2}{17}\).

This result means there is a \(\frac{2}{17}\) chance, or probability, for each attempt to be successful.
Probability calculation is a skill used extensively in fields like gambling, finance, and decision-making, as it provides a clear expectation of outcomes.

Cross-Multiplication

Cross-multiplication is a straightforward algebraic method used to simplify equations that involve fractions.

Imagine you begin with a ratio or a fraction: \(\frac{n}{m} = \frac{P(A)}{1-P(A)}\). To solve this fraction equation, you use cross-multiplication to eliminate the fractions and simplify the equation. Here’s how you do it:

• Perform the cross-multiplication: Multiply across the equals sign to get \(n(1-P(A)) = mP(A)\).
• Distribute and rearrange terms: Expand and combine all terms involving \(P(A)\) on one side: \(n - nP(A) = mP(A)\).
• Factor \(P(A)\) out: Rearrange to get \(n = P(A)(n + m)\), which simplifies to finding \(P(A) = \frac{n}{n + m}\).

Cross-multiplication is crucial in many math problems, especially those dealing with proportions, ratios, and probability, as it makes them much more manageable to solve.