Problem 38 - Algebra for College Students

Question

What are the odds in favor of getting at least one 6 when tossing a pair of dice?

Step-by-Step Solution

Answer

The odds in favor are \(11:25\).

Step 1: Determine Total Possible Outcomes

When tossing a pair of dice, each die has 6 faces. So, there are a total of \(6 \times 6 = 36\) possible outcomes.

Step 2: Calculate the Complementary Event

Calculate the number of outcomes where no sixes appear. For each die, there are 5 outcomes that are not a six. Therefore, the number of outcomes where neither die shows a 6 is \(5 \times 5 = 25\).

Step 3: Calculate Outcomes with at Least One 6

Subtract the number of outcomes with no sixes from the total outcomes: \(36 - 25 = 11\).

Step 4: Calculate Odds in Favor

The odds in favor of an event are given by the ratio of the number of favorable outcomes to the number of unfavorable outcomes. Here, favorable outcomes are 11 and unfavorable outcomes are 25. So, the odds in favor are \(11:25\).

Key Concepts

Complementary Events

A complementary event involves scenarios where the outcome is opposite to what you're interested in. When dealing with dice, if we want to avoid rolling a 6, we're interested in the complementary event of 'no 6s appearing.'
This simplifies probability calculations.
Complements make calculations easier. When we find the probability of the complementary event, we can subtract that from 1 to get our desired probability.
Here’s a simple example:
* If the chance of rain tomorrow is 30% (event), then the chance it won't rain (complementary event) is 70%.

For two dice, if each has a 1 in 6 chance of rolling a 6, then the probability that neither dice shows a 6 is simpler to find by working with complements first.

Remember these:
* Probability of event and its complement adds up to 1.
* Using complementary events helps in handling complex situations more easily.

Total Possible Outcomes

Total possible outcomes represent all the different results one can get from an experiment. In our dice example, each die has 6 faces, leading to many possible combinations when you roll two dice together.
To find total outcomes when rolling two dice, you multiply the number of faces on the first die by the faces on the second: \(6 \times 6 = 36\)So, there are 36 possible results.

Why is this useful?

• Knowing the total outcomes helps calculate the probabilities of specific events.


• It provides a foundation to figure out both favorable events (desired outcomes) and unfavorable events (the rest).

Always start by understanding the complete set of outcomes to simplify further calculations.

Calculating Odds

Odds help compare the likelihood of an event occurring to it not occurring. This is different from probability, which measures the chance of an event happening.

To calculate odds in favor, you need:

• The number of favorable outcomes
• The number of unfavorable outcomes

In our dice example:

• There are 11 favorable outcomes (at least one 6)
• 25 unfavorable outcomes (no 6s)

So, the odds in favor are given by the ratio 11:25.

This tells us that for every 11 favorable results, there are 25 unfavorable results.
Always remember:

• Odds are represented as favorable:unfavorable.
• They offer a different perspective from probability but are rooted in the same calculations.

By breaking down the problem into complementary events, total outcomes, and calculating odds, solving any probability question becomes manageable.