Question

the probability that event $a$ will occur is $p(a)=0.72$. what is the probability (in decimal form) that event $a$ will not occur? $p(overline{a}) =$ what are the odds for event $a?$ to what are the odds against event $a?$ to simplify your answers.

Explanation:

Step1: Recall probability formula

The probability of an event $A$ and its complement $\overline{A}$ satisfy $P(A)+P(\overline{A}) = 1$.

Step2: Calculate $P(\overline{A})$

Given $P(A)=0.72$, then $P(\overline{A})=1 - P(A)=1 - 0.72 = 0.28$.

Step3: Calculate odds for event $A$

The odds for an event $A$ is given by $\frac{P(A)}{P(\overline{A})}$. Substituting the values, we have $\frac{0.72}{0.28}=\frac{72}{28}=\frac{18}{7}$, so the odds for event $A$ is 18 to 7.

Step4: Calculate odds against event $A$

The odds against an event $A$ is given by $\frac{P(\overline{A})}{P(A)}$. Substituting the values, we get $\frac{0.28}{0.72}=\frac{28}{72}=\frac{7}{18}$, so the odds against event $A$ is 7 to 18.

Answer:

$P(\overline{A}) = 0.28$
Odds for event $A$: 18 to 7
Odds against event $A$: 7 to 18