Question

in gambling, the chances of winning are often written in terms of odds rather than probabilities. the odds of winning is the ratio of the number of successful outcomes to the number of unsuccessful outcomes. the odds of losing is the ratio of the number of unsuccessful outcomes to the number of successful outcomes. for example, if the number of successful outcomes is 2 and the number of unsuccessful outcomes is 3, the odds of winning are 2:3 (read \2 to 3\) or \\(\frac{2}{3}\\). also, if the odds of winning are \\(\frac{2}{3}\\), the probability of success is \\(\frac{2}{5}\\).
the odds of an event occurring are 1:2. find (a) the probability that the event will occur and (b) the probability that the event will not occur.

(a) the probability that the event will occur is \\(square\\)
(type an integer or decimal rounded to the nearest thousandth as needed.)

(b) the probability that the event will not occur is \\(square\\)
(type an integer or decimal rounded to the nearest thousandth as needed.)

Explanation:

Step1: Understand Odds to Probability

Odds of winning (event occurring) are \( \frac{\text{successful outcomes}}{\text{unsuccessful outcomes}} = \frac{1}{2} \). Probability of event occurring is \( \frac{\text{successful}}{\text{successful + unsuccessful}} \).
So, successful = 1, unsuccessful = 2. Total outcomes = \( 1 + 2 = 3 \). Probability (occur) = \( \frac{1}{3} \approx 0.333 \).

Step2: Probability of Not Occurring

Probability (not occur) = \( 1 - \) Probability (occur) = \( 1 - \frac{1}{3} = \frac{2}{3} \approx 0.667 \).

Answer:

(a) \( 0.333 \)
(b) \( 0.667 \)