Solution:
Given:
The table of the poll results;
The formula for calculating probability is given by;
[tex]\text{Probability}=\frac{n\text{ umber of required outcomes}}{n\text{ umber of total outcomes}}[/tex]
a) The P(Democrat) is;
[tex]\begin{gathered} \text{P(Democrat)}=\frac{\text{ total number of democrat}}{n\text{ umber of total outcomes}} \\ P(\text{Democrat)}=\frac{236}{651} \\ P(\text{Democrat)}=0.3625 \end{gathered}[/tex]
Therefore, the probability that it is a democrat is 0.3625
b) The P(worse) is;
[tex]\begin{gathered} \text{P(worse)}=\frac{\text{ total number of worse}}{n\text{ umber of total outcomes}} \\ P(\text{worse)}=\frac{299}{651} \\ P(\text{worse)}=0.4593 \end{gathered}[/tex]
Therefore, the probability that it is worse is 0.4593
c) The P(worse|Democrat)- This is a conditional probability.
We use the conditional probability formula to solve this.
Conditional probability is given by;
[tex]P(A|B)=\frac{P(A\cap B)}{P(B)}[/tex]
Hence,
[tex]\begin{gathered} \text{Probability of worse given democrat=}\frac{P(worse\text{ and democrat)}}{\text{Probability (democrat)}} \\ P(\text{worse}|\text{democrat)}=\frac{137}{651}\text{ /}\frac{236}{651} \\ P(\text{worse}|\text{democrat)}=\frac{137}{651}\times\frac{651}{236} \\ P(\text{worse}|\text{democrat)}=\frac{137}{236} \\ P(\text{worse}|\text{democrat)}=0.5805 \end{gathered}[/tex]
Therefore, the probability of worse given democrat is 0.5805
d) The P(Democrat|worse) is;
[tex]\begin{gathered} \text{Probability of democrat given worse=}\frac{P(democrat\text{ and worse)}}{\text{Probability (worse)}} \\ P(\text{democrat}|\text{worse)}=\frac{137}{651}\text{ /}\frac{299}{651} \\ P(\text{democrat}|\text{worse)}=\frac{137}{651}\times\frac{651}{299} \\ P(\text{democrat}|\text{worse)}=\frac{137}{299} \\ P(\text{democrat}|\text{worse)}=0.4582 \end{gathered}[/tex]
Therefore, the probability of democrat given worse is 0.4582
e) The P(Democrat and worse) is;
[tex]\begin{gathered} P(A|B)=\frac{P(A\cap B)}{P(B)} \\ P(A\cap B)=P(A|B)\times P(B) \\ P(\text{Democrat and worse)=P(Democrat|worse)}\times P(worse) \\ P(\text{Democrat}|\text{worse) calculated in option (d) above = 0.4582} \\ P(\text{worse) calculated in option (b) above = 0.4593} \\ \\ \text{Hence,} \\ P(\text{Democrat and worse)=P(Democrat|worse)}\times P(worse) \\ P(\text{Democrat and worse)=}0.4582\times0.4593 \\ P(\text{Democrat and worse)=}0.2105 \\ \\ \\ \\ \text{Alternatively,} \\ P(\text{Democrat and worse)=}\frac{n\text{ umber of democrat and worse}}{\text{total number}} \\ P(\text{Democrat and worse)=}\frac{137}{651} \\ P(\text{Democrat and worse)=}0.210445 \\ P(\text{Democrat and worse)=}0.2105 \end{gathered}[/tex]
Therefore, the probability of democrat and worse is 0.2105
