Sure, let's go through the solution step-by-step.
### Q6a
To determine the probability that a randomly selected teenager is female and their favorite hobby is watching films, we start with the relevant counts:
- There are 8 females whose favorite hobby is watching films.
- The total number of teenagers is 32.
The probability of selecting a female whose favorite hobby is watching films is given by the ratio of these numbers:
[tex]\[
\text{Probability} = \frac{\text{Number of females whose favorite hobby is watching films}}{\text{Total number of teenagers}} = \frac{8}{32}
\][/tex]
To simplify this fraction:
[tex]\[
\frac{8}{32} = \frac{1}{4}
\][/tex]
So, the probability that a teenager selected at random is female and their favorite hobby is watching films is:
[tex]\[
\frac{1}{4}
\][/tex]
### Q6b
Now, let's address the next part of the question. After the teenager from part (a) rejoins the group, another teenager is picked at random. We need to find the probability that this new selected teenager is male and their favorite hobby is playing video games.
- There are 2 males whose favorite hobby is playing video games.
- However, given that the teenager from part (a) rejoined the group, the total number of teenagers is reduced by 1 initially, making it 31 (but it becomes 32 again since they rejoined).
The probability of selecting a male whose favorite hobby is playing video games now is:
[tex]\[
\text{Probability} = \frac{\text{Number of males whose favorite hobby is playing video games}}{\text{Total number of teenagers left}} = \frac{2}{31}
\][/tex]
So the probability, now after the first teenager rejoins, that another randomly selected teenager is male and their favorite hobby is playing video games, is:
[tex]\[
\frac{2}{31}
\][/tex]
In summary:
- Q6a: The probability is [tex]\(\frac{1}{4}\)[/tex].
- Q6b: The probability after the rejoin is [tex]\(\frac{2}{31}\)[/tex].
