The probability that a vehicle entering the Luray Caverns has Canadian

TokNeekCepTdh 2021-11-20 Answered
The probability that a vehicle entering the Luray Caverns has Canadian license plates is 0.19; the probability that it is a camper is 0.38; and the probability that it is a camper with Canadian license plates is 0.13. What is the probability that
(a) a camper entering the Luray Caverns has Canadian license plates?
(b) a vehicle with Canadian license plates entering the Luray Caverns is not a camper?
(c) a vehicle entering the Luray Caverns does not have Canadian plates or is not a camper?

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Expert Answer

Symbee
Answered 2021-11-21 Author has 1049 answers
Step 1
Let the event that a vehicle entering the Luray Caverns has Canadian license plates is "A".
Let the event that a vehicle entering the Luray Caverns is a camper is denoted by "B".
Let the event that a vehicle entering the Luray Caverns is a camper with Canadian license plates is denoted by \(\displaystyle{\left({A}\cap{B}\right)}\).
So,
\(\displaystyle{P}{\left({A}\right)}={0.19}\)
\(\displaystyle{P}{\left({B}\right)}={0.38}\)
\(\displaystyle{P}{\left({A}\cap{B}\right)}={0.13}\)
Step 2
(a) We find the probability that a camper entering the Luray Caverns has Canadian license plates as,
\(\displaystyle{\left({A}{\mid}{B}\right)}={\frac{{{P}{\left({A}\cap{B}\right)}}}{{{P}{\left({B}\right)}}}}\)
\(\displaystyle={\frac{{{0.13}}}{{{0.38}}}}\)
\(\displaystyle={0.3421}\)
Hence, required probability is 0.3421.
(b) We find the probability that a vehicle with Canadian license plates entering the Luray Caverns is not a camper as,
\(\displaystyle{P}{\left({B}^{{{C}}}{\mid}{A}\right)}={\frac{{{P}{\left({B}^{{{C}}}\cap{A}\right)}}}{{{P}{\left({A}\right)}}}}\)
\(\displaystyle={\frac{{{P}{\left({A}\right)}-{P}{\left({A}\cap{B}\right)}}}{{{P}{\left({A}\right)}}}}\) [By formula, \(\displaystyle{P}{\left({A}\right)}={P}{\left({A}\cap{B}\right)}+{P}{\left({B}^{{{C}}}\cap{A}\right)}\)]
\(\displaystyle={\frac{{{0.19}-{0.13}}}{{{0.19}}}}\)
\(\displaystyle={0.3158}\)
Hence, required probability is 0.3158.
(c) We find the probability that a vehicle entering the Luray Caverns does not have Canadian plates or is not a camper is,
\(\displaystyle{P}{\left({A}^{{{C}}}\cup{B}^{{{C}}}\right)}={P}{\left({A}\cap{B}\right)}^{{{C}}}\) [By formula, \(\displaystyle{P}{\left({A}^{{{C}}}\cup{B}^{{{C}}}\right)}={P}{\left({A}\cap{B}\right)}^{{{C}}}\)]
\(\displaystyle={1}-{P}{\left({A}\cap{B}\right)}\)
\(\displaystyle={1}-{0.13}\)
\(\displaystyle={0.87}\)
Hence, required probability is 0.87.
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