+2440 Before we start, let's define a few terms.
Odds is the ratio of the probability that an event can occur to the probability that an event cannot occur. Here is the formula in algabraic notation, if it makes it easier to understand.
\(\text{Odds}=\frac{p}{1-p}\)
p = probability
Probability is the ratio of the ways an event suits a condition to the number of possible outcomes.
\(p=\frac{\text{# of events}}{\text{# of outcomes}}\)
p = probability
Knowing this, we can answer all the questions up to 11.
8)
There are 3 primary colors in the colored cube, and this allows us to calculate the odds.
\(\text{Odds}=\frac{{\frac{1}{2}}}{1-\frac{1}{2}}=\frac{\frac{1}{2}}{\frac{1}{2}}=1:1\)
9)
Let's calculate the probability of not rolling a blue or purple. If we wanted to roll a blue or purple, the probability would be \(\frac{2}{6}=\frac{1}{3}\). The probability of that event not occuring is \(1-p\). Therefore, \(1-\frac{1}{3}=\frac{3}{3}-\frac{1}{3}=\frac{2}{3}\). Now, let's determine the odds.
| \(\text{Odds}=\frac{\frac{2}{3}}{1-\frac{2}{3}}\) | Simplify the denominator |
| \(1-\frac{2}{3}=\frac{3}{3}-\frac{2}{3}=\frac{1}{3}\) | |
| \(\text{Odds}=\frac{\frac{2}{3}}{\frac{1}{3}}\) | Multiply by \(\frac{3}{3}\) to eliminate the ugly denominator. |
| \(\text{Odds}=\frac{2}{3}*\frac{3}{1}=\frac{6}{3}=2:1\) | |
10)
There are 3 primary colors on the cube, which means that the probability is \(\frac{3}{6}=\frac{1}{2}\)
11) The probability of an event not occuring is represented by \(1-p\), so this means that \(1-\frac{1}{6}\) is the probability. Now, just simplify this.
\(1-\frac{1}{6}=\frac{6}{6}-\frac{5}{6}=\frac{1}{6}\)
