Probability

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tomato

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May 12, 2005
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Here is my problem:

A person decides to take a vacation and has three destinations in mind: Las Vegas, Bahamas, and Hawaii. She is twice as likely to go to Hawaii as to Las Vegas, and three times as likely to go to Las Vegas as to the Bahamas. Find the probability that she vacations in Las Vegas.

*I really do not know how to approach this problem. A friend of mine got .3 or 30% as the answer was but I am not sure if that is correct or how she even came to that conclusion. Any help would be great! Thanks ;)
 
\(\displaystyle P_{H} = 2P_{L}\)

\(\displaystyle P_{L} = 3P_{B}\)

So,

\(\displaystyle P_{H} = 6P_B\)
\(\displaystyle P_{L} = 3P_B\)
\(\displaystyle P_{B} = P_B\)

\(\displaystyle \sum P = 1 = P_B+P_H+P_L\)

\(\displaystyle 1 = 10P_B\)

\(\displaystyle P_B = 0.1\)

\(\displaystyle P_H = 6(0.1) = 0.6\)

\(\displaystyle P_L = 3(0.1) = 0.3\)
 
Hello, tomato!

daveyp's solution and explanation are absolutely correct.
Here's my baby-talk version . . .

A person decides to take a vacation and has three destinations in mind: Las Vegas, Bahamas, and Hawaii.
She is twice as likely to go to Hawaii as to Las Vegas, and three times as likely to go to Las Vegas as to the Bahamas.
Find the probability that she vacations in Las Vegas.
Click to expand...
Think of the three probabilities are "pieces of pie".
Let \(\displaystyle H\) = Hawaii, \(\displaystyle L\) = Las Vegas, \(\displaystyle B\) = Bahamas.

We are told that: \(\displaystyle H \,=\,2L\;\) (\(\displaystyle H\) gets twice as much pie as \(\displaystyle L\).)
\(\displaystyle \;\;\)also: \(\displaystyle L\,=\,3B\;\) (\(\displaystyle L\) gets three times as much pie as \(\displaystyle B\).)

Hence, \(\displaystyle H\,=\,2L\,=\,2(3B)\,=\,6B\;\) (\(\displaystyle H\) gets six times as much pie as \(\displaystyle B\).)

The pie is divided into the proportion: \(\displaystyle \,H:L:B\,=\,6:3:1\)


So the pie is divided into ten equal parts.

\(\displaystyle \;\;H\) gets 6 pieces: \(\displaystyle \,P(H)\,=\,\frac{6}{10}\.=\,0.6\)

\(\displaystyle \;\;L\) gets 3 pieces: \(\displaystyle \,P(L)\,=\,\frac{3}{10}\,=\,0.3\)

\(\displaystyle \;\;B\) gets 1 piece: \(\displaystyle \;P(B)\,=\,\frac{1}{10}\,=\,0.1\)
 
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