Question

$1994$, the U.S. government held a lottery to issue $55000$Green Cards (permits for non-citizens to work legally in the U.S.). Renate Deutsch, from Germany, was one of approximately $6.5$million people who entered this lottery. Let G = won green card.

a. What was Renate’s chance of winning a Green Card? Write your answer as a probability statement.

b. In the summer of $1994$, Renate received a letter stating she was one of $110,000$finalists chosen. Once the finalists were chosen, assuming that each finalist had an equal chance to win, what was Renate’s chance of winning a Green Card? Write your answer as a conditional probability statement. Let F = was a finalist.

c. Are G andF independent or dependent events? Justify your answer numerically and also explain why.

d. Are G and F mutually exclusive events? Justify your answer numerically and explain why.

Short answer

(a) $P\left(G\right)=0.008$.

(b) $P\left(G|F\right)=0.5$.

(c) $P(G\cap F)\ne P\left(G\right)P\left(F\right)$.

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

Step-by-step solution

Given information (part a)

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.

Explanation (part a)

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$P\left(G\right)=0.008$.

Given information (part b)

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.

Explanation (part b)

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$P\left(G/F\right)=0.5$.

Given information (part c)

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.

Explanation (part c)

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

Test the independency by,

$$\begin{gathered} P(G\cap F)=P(G/F)P\left(F\right) \\ P(G\cap F)=1.8181\times {10}^{-5}\times 9.09091\times {10}^{-6} \\ P(G\cap F)=1.6528\times {10}^{-10} \\ P(G\cap F)\ne 4.5454\times {10}^{-6} \end{gathered}$$

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

Given information (part d)

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.

Explanation (part d)

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