Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Suppose the probability that it will rain tomorrow is . 3 . a. What are the odds that it will rain tomorrow? b. What are the odds that it will not rain tomorrow?

Step-by-Step Solution

Answer

a. The odds of rain tomorrow are approximately \(0.4286\), which can be expressed as a ratio of \(3:7\) (3 chances of rain to 7 chances of not rain). b. The odds of not raining tomorrow are approximately \(2.3333\), which can be expressed as a ratio of \(7:3\) (7 chances of not rain to 3 chances of rain).

Step 1: Find the probability of Event A (rain)

The probability of rain tomorrow is given as 0.3. So, we have: P(A) = 0.3

Step 2: Find the probability of Event B (not rain)

Since there are only two possible outcomes (rain and not rain), the probability of not raining is the complement of the probability of rain: P(B) = 1 - P(A) = 1 - 0.3 = 0.7

Step 3: Calculate the odds of Event A (rain)

To find the odds of rain tomorrow, we compute the ratio of the probability of rain to the probability of not rain: Odds(A) = P(A) / P(B) = 0.3 / 0.7 ≈ 0.4286

Step 4: Calculate the odds of Event B (not rain)

To find the odds of not raining tomorrow, we compute the ratio of the probability of not rain to the probability of rain: Odds(B) = P(B) / P(A) = 0.7 / 0.3 ≈ 2.3333

Step 5: Express odds as ratios

a. The odds of rain tomorrow are approximately 0.4286, which can be expressed as a ratio of 3:7 (3 chances of rain to 7 chances of not rain). b. The odds of not raining tomorrow are approximately 2.3333, which can be expressed as a ratio of 7:3 (7 chances of not rain to 3 chances of rain).

Key Concepts

Odds Calculation

Calculating odds involves finding the likelihood of an event occurring in comparison to it not occurring. In probability theory, the odds for an event can be determined using probabilities. Suppose you know that the probability of rain tomorrow is 0.3. To find the odds that it will rain, you compare the probability of rain to the probability of it not raining, which is known as the complementary event.
For instance, if the probability of rain (Event A) is 0.3, you first identify the probability that it will not rain (Event B), which is 0.7. The odds formula is straightforward:

• Odds of Event A = Probability of Event A / Probability of Event B
• Using our probabilities: Odds(A) = 0.3 / 0.7 ≈ 0.4286

This means for every occurrence of rain, there are approximately 2.333 occurrences of no rain. In simpler terms, you can express this in a ratio of chances: 3:7 (3 chances of rain for every 7 chances of no rain).

Complementary Events

In probability, an event and its complement together account for all possible outcomes. If an event is about it raining tomorrow, the complementary event is it not raining. Since these are exhaustive outcomes, their probabilities add up to 1. This relationship is crucial in odds calculation.
Take the probability that it will rain (P(A)) which is given as 0.3. The complement is the probability it will not rain (P(B)), calculated by subtracting P(A) from 1:

• P(B) = 1 - P(A) = 1 - 0.3 = 0.7

This complementary nature simplifies many calculations. For instance, using this probability, one can easily swap between calculating odds for rain and no rain, depending on which assessment you want to make. Once you know one probability, the other follows easily, demonstrating the completeness of complementary events in probability theory.

Ratio of Probabilities

The ratio of probabilities is a key concept for determining odds. It shows how one outcome compares to another in terms of likelihood. For example, you have the probabilities of rain 0.3 and no rain 0.7, respectively. To ascertain the odds, you need only take one probability and divide it by the other.
When calculating the odds of rain (Odds A), take the probability of rain divided by no rain:

• Odds(A) = 0.3 / 0.7 ≈ 0.4286

Similarly, to find odds of no rain (Odds B), reverse the order:

• Odds(B) = 0.7 / 0.3 ≈ 2.3333

These odds provide a clear numerical ratio of how likely one event is in relation to its complement. In practical terms, these ratios translate into more digestible forms like 3:7 or 7:3, facilitating easier interpretation of the likelihood of events in daily scenarios.