91Ó°ÊÓ

The total number of Green Cards issued by the U.S.government is $55000$and Renate Deutsch is among $6.5$million people who entered this lottery.

Probability Renate's chance of winning a green card is,

$$\begin{gathered} P\left(G\right)=\frac{55000}{6500000} \\ P\left(G\right)=0.008 \end{gathered}$$

Therefore, the required probability is$P\left(G\right)=0.008$.

The total number of Green Cards issued by the U.S. government is $55,000$ and Renate Deutsch is among $6.5$ million people who entered this lottery.

Conditional probability Renate's chance of winning a green card(G) is,

$$\begin{gathered} P\left(\text{Finalist win}\right(F\left)\right)=\frac{1}{110000} \\ P(G/F)=\frac{55000}{110000} \\ P\left(G/F\right)=0.5 \end{gathered}$$

Therefore required conditional probability is$P\left(G/F\right)=0.5$.

The total number of Green Cards issued by theU.S. government is $55,000$ and Renate Deutsch is among $6.5$ million people who entered this lottery.

$$\begin{gathered} P\left(G\right)=\frac{1}{55000} \\ P\left(G\right)=1.8181×{10}^{-5} \\ P\left(F\right)=\frac{1}{110000} \\ P\left(G\right)=9.0909×{10}^{-6} \\ \mathrm{P}(\mathrm{G}/\mathrm{F})=\frac{\mathrm{P}(\mathrm{G}âˆ\copyright \mathrm{F})}{\mathrm{P}\left(\mathrm{F}\right)} \\ P(Gâˆ\copyright F)=P(G/F)P\left(F\right) \\ P(Gâˆ\copyright F)=0.5×9.09091×{10}^{-6} \\ \mathrm{P}(\mathrm{G}âˆ\copyright \mathrm{F})=4.5454×{10}^{-6} \end{gathered}$$

Test the independency by,

$$\begin{gathered} P(Gâˆ\copyright F)=P(G/F)P\left(F\right) \\ P(Gâˆ\copyright F)=1.8181×{10}^{-5}×9.09091×{10}^{-6} \\ P(Gâˆ\copyright F)=1.6528×{10}^{-10} \\ P(Gâˆ\copyright F)≠4.5454×{10}^{-6} \end{gathered}$$

These events G and F are dependent, because $P(Gâˆ\copyright F)≠P\left(G\right)P\left(F\right)$. She has a better chance of winning a green card once she becomes a finalist.

The total number of Green Cards issued by the U.S. government is $55000$ and Renate Deutsch is among $6.5$ million people who entered this lottery.

No, each winner comes out of the pool of finalists, so$P\left(G\text{andFinalist}\right)$doesn't equal$0$..