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}\\). (note: if the odds of winning are \\(\frac{2}{3}\\), the probability of success is \\(\frac{2}{5}\\).)

the odds of an event occurring are 4:6. 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 .
(type an integer or decimal rounded to the nearest thousandth as needed.)

Explanation:

Define the relationship between odds and probability

The odds of an event occurring are given as the ratio of the number of successful outcomes to the number of unsuccessful outcomes. If the odds of an event occurring are \(a:b\), this means there are \(a\) successful outcomes and \(b\) unsuccessful outcomes.

The total number of possible outcomes is:

$$ \text{Total outcomes} = a + b $$

The probability \(P(E)\) that the event will occur is the ratio of successful outcomes to the total outcomes:

$$ P(E) = \frac{a}{a + b} $$

The probability \(P(E')\) that the event will not occur is the ratio of unsuccessful outcomes to the total outcomes:

$$ P(E') = \frac{b}{a + b} $$

Identify the given values

We are given that the odds of the event occurring are \(4:6\).

• Number of successful outcomes, \(a = 4\)
• Number of unsuccessful outcomes, \(b = 6\)

Calculate the total number of outcomes

Using the values of \(a\) and \(b\):

$$ \text{Total outcomes} = 4 + 6 = 10 $$

Calculate the probability that the event will occur

Using the formula for the probability of occurrence:

$$ P(E) = \frac{4}{10} = 0.4 $$

Calculate the probability that the event will not occur

Using the formula for the probability of non-occurrence:

$$ P(E') = \frac{6}{10} = 0.6 $$

Answer:

Question 1

(a) The probability that the event will occur is <blank>0.4</blank>.

Question 2

(b) The probability that the event will not occur is <blank>0.6</blank>.