Let's address each part of the problem step-by-step.
### Part (a) Finding the odds in favor of receiving a gift
1. Identify the probability of receiving a gift:
The probability of receiving a gift is given as [tex]\(\frac{5}{18}\)[/tex].
2. Calculate the probability of not receiving a gift:
If the probability of receiving a gift is [tex]\(\frac{5}{18}\)[/tex], then the probability of not receiving a gift is:
[tex]\[
1 - \frac{5}{18} = \frac{18 - 5}{18} = \frac{13}{18}
\][/tex]
3. Determine the odds in favor of receiving a gift:
Odds in favor of an event is the ratio of the probability of the event occurring to the probability of the event not occurring.
[tex]\[
\text{Odds in favor of receiving a gift} = \frac{\frac{5}{18}}{\frac{13}{18}} = \frac{5}{13}
\][/tex]
Therefore, the odds in favor of receiving a gift are [tex]\( \frac{5}{13} \)[/tex], which numerically is:
[tex]\[
0.38461538461538464
\][/tex]
### Part (b) Finding the probability of the box having a toy given the odds against it
1. Identify the odds against the box having a toy:
The question states that the odds against the box having a toy are [tex]\(\frac{4}{5}\)[/tex].
2. Understand the relationship between odds and probability:
The odds against an event is the ratio of the probability of the event not occurring to the probability of the event occurring. Here it is [tex]\(\frac{4}{5}\)[/tex].
3. Express the given information in terms of probability:
Let [tex]\( p \)[/tex] be the probability of the box having a toy. The odds against having a toy means:
[tex]\[
\frac{P(\text{no toy})}{P(\text{toy})} = \frac{4}{5}
\][/tex]
Since [tex]\( P(\text{no toy}) = 1 - p \)[/tex], we can write:
[tex]\[
\frac{1 - p}{p} = \frac{4}{5}
\][/tex]
4. Solve for [tex]\( p \)[/tex]:
To find [tex]\( p \)[/tex], we solve the equation:
[tex]\[
1 - p = \frac{4}{5}p
\][/tex]
Multiplying both sides by 5 to clear the fraction:
[tex]\[
5(1 - p) = 4p
\][/tex]
[tex]\[
5 - 5p = 4p
\][/tex]
[tex]\[
5 = 9p
\][/tex]
Therefore,
[tex]\[
p = \frac{5}{9}
\][/tex]
So, the probability of the box having a toy is:
[tex]\[
0.5555555555555556
\][/tex]
By following these steps carefully, we have found:
(a) The odds in favor of receiving a gift are [tex]\( \frac{5}{13} \)[/tex] or approximately 0.3846.
(b) The probability of the box having a toy is [tex]\( \frac{5}{9} \)[/tex] or approximately 0.5556.
